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THE 

FIELD ENGINEER: 

A 

IHantis 33oolt of Practice 

IN THE 

URVEY, LOCATION, AND TRACK-WORK OF 

RAILROADS ; 

COXTAIXISTG 

A LAEGE COLLECTION Of EULES AND TABLES, 

ORIGINAL AND SELECTED, 



APPLICABLE TO BOTH THE STANDARD AND THE 
NARROW GAUGE. 



AND PREPARED WITH SPECIAL REFERENCE TO 
THE WANTS OF 



THE YOUNG 'ENGLs'EEIi. 



WILLIAM FINDLAYSHUNK, C.E. 
FIFTEENTH EDITION, EE VISED AND ENLAEGED. 



NEW YORK: 

D. VAN NOSTRAND COMPANY, 

23 MUEBAY AND 27 WaKREN STRBBTS. 

1903. 



THE LIBRARY @F 
CONGRESS, 

Two Copies Received 

APR f 1903 

Copyright Entry 
CLASS CC )»<c.No. 
COPY B. 



1 






F-S.O 

as 



Copyright, 1890, 1903, 

bt d. van nostrand company. 




THE AUTHOR 

Affectionately Dedicates tbfs JBooft 

TO 

ALBERT J. SCHERZER, C.E., 

§)tb ^omrabe an5 pear ^rienb, 

IN TOKEN OF ESTEEM FOR HIS PROFESSIONAL 
ATTAINMENTS AND RESPECT FOR HIS 
MANLY CHARACTER 



PEEFAOE. 



The author's principal aim in preparing this volume has 
been, as its title indicates, to serve that large class of young 
engineers who, like himself, have not had the advantage of a 
technical education before going out for their livelihood. 

The initial chapters are, therefore, given to a compendious 
exposition of those mathematical truths and methods which 
they must needs become familiar with from the beginning. 
Plane Trigonometry, Logarithms, and propositions relating to 
the circle, are tools of the ci-aft in constant use; ready han- 
dling of them is an indispensable condition of excellence. Be 
not discouraged by obscurities and difficulties at the outset; 
light will gradually break on the scrutinizing eye, and a way 
always open to manful effort. 

These chapters are followed by instructions as to the adjust- 
ment and use of instruments, and hints concerning field rou- 
tine, which it is thought will be found acceptable to the 
inexperienced learner. The same may be said of the articles 
on staking out work, and those on track problems, with which 
the text of the book closes. They have been written with the 
author's own early ignorance in mind, and with a wish to set 
the subjects forth as plainly as possible, disembarrassed of 
hard words in the description, and of unpractical niceties in 
the operation. 

The chapter on field location is believed to include all the 
problems likely to occur. The author, in compiling it, has 
taken those only which have arisen in his own practice, and 
which, therefore, may arise in the practice of others. His 



PREFACE. 



own practice having been unusually large and diversified 
probably the examples given will prove adequate, directly o; 
indirectly, to all contingencies. 

No attempt has been made to swell the bulk of the volum 
with imaginary cases ; the object being, not to provide barre: 
mathematical exercises, but to teach useful knowledge. 

Problems, also, affecting location in its economical aspects, 
— the balancing of physical and financial conditions, equatin 
of alternative lines, and the like, — do not come within th 
scope of the work, and are therefore not treated. 

Considerable pains have been spent on the tables. Iloweve 
far the j^oung engineer may eventually outgo his teacher as re 
gards the text of the b3ok, these are implements of his a 
which never become antiquated, and can never fall into dis 
use. Those herein contained which are original will, it ii 
hoped, be esteemed worthy of place with their well-approved 
associates. 

The author invites friendly criticism : he would be pleased 
\i receive suggestions, both for the improvement of the book, 
and for the correction of possible errors in it, should another 
edition be called for. 

In dismissing the work from his hands, the precarious 
snatches of time occupied in its preparation, by day and by 
night, during the past two years, which might have been more 
agreeably spent in reading, talking, or musing, recur to the 
writer's mind; and the thought arises. To what end or from 
what motive do people undertake these technical labors? Why 
should Forney and Bourne toil to simplify steam for our ap- 
prehension; Nystrom to compile mechanical, Moles worth and 
Trautwine to epitomize civil engineering; Henck to prepare 
his elegant manual of field mathematics; Box to illustrate 
hydraulics; and Shreve, with lucid pen, to make clear for us 
the strains in truss or arch? The ordinary motives to en- 
deavor here have no place. There is neither fame noi- profit 
in these drudging enterprises. At best the author gives name 






PREFACE. Vii 

to his book; he remains impersonal, — known but indirectly, 
and but to a class. How, then, shall we account for his labors? 
I take it, the Father of mankind has not only made our minds 
to hunger for knowledge as our bodies for food, but has also 
imposed upon u* a kindlv law of communion, by virtue whereof 
we cannot do otherwise, without violence to generous nature, 
than share with our fellows whatsoever we have learned that 
seems new and useful. Under this law these beneficial works 
would appear to have had their being, and thus pure are they 
from the stain of selfishness. 

Though the present writer would not arrogate equal fellow- 
ship in the eminent brotherhood named, yet he may justly 
claim like pureness from unworthy motive, and certainly feels 
hke comfort at heart to that which they must know, for having 
discharged, in what measure it has been laid upon him, the 

divine obligation. 

WM. F. SHUNK. 
Rahwat, N J. 



PEEFACE TO THE NINTH EDITION. 



Although the writer has thanked individually all who 
have sent him errata from time to time, he cannot regard this 
enlarged edition of the "Field Engineer" as complete with- 
out a general salutation to them. Thanks, too, for pleasing 
evidences, received from many hands, that the book has been 
helpful to those for whom it was intended. 

He would make special acknowledgment to Mr. T. C. 
Mendenhall, Supt. U. S. Coast and Geodetic Survey, and to 
his assistant, Mr. C. A. Schott, for the new astronomical tables 
herein first printed, which were kindly computed for this 
edition in response to a call for much less. 

He would also express his obligation to Mr. Fred. Brooks^ 
C.E., Boston, Mass., whose critical suggestions for the better 
ment of the text have m the main been adopted. 

These gentlemen are strangers to their beneficiary. He 



Viii PREFACE. 

knows them only by good report, and by these personal 
courtesies. He had no claim on them but that of Saint Paul's 
"one blood." He can pay back thanks alone, — a residual 
debt being left over to do the like by others as they have done 
by him, to the extent of his limited ability. Thus good turns 
go round, and we civilize by mutual service. 

W. F. S. 
Harrisburg, Pa., 
March, 1890. 



ABBEEYIATIONS. 



-\- Increased by. 

— Diminished by. 

X Multiplied by. 

-i- Divided by. 

= Equal to. 

•. • Since, or seeing tliat. 

. '. Hence, or therefore. 

: Indicates the quotient of one divided by the other of the 
quantities it connects, called sometimes the ratio of the quan- 
tities. 

: : Indicates an equality of ratios, and connects equal ratios 
In 2i proportion. Thus, a : h '.'. c : d indicates that a -f- 6 = c 
■^ fZ; or it may be read, a is to 6 as c is to d. 

{ ) Brackets indicate that the operations embraced by them 
shall first be performed, and the result treated as a single term 
lu the remaining processes required by a formula. Thus, 
aXb -^{a-\-b) requires that the product of a and h shall be 
divided by their sum. This expression may also be written 

— —., or a X — ri;' 01" & -\ r^i- If the brackets be omitted 

a-\-b a-\- b a-\-b 

the expression a xb -^ a -\-b w^ould mean \-b. 

d 

A?. A small secondary figure annexed thus to an expression 
is called its exponent. It requires the principal to which it is 
attached to be used as many times in continued multiplication 
as there are units in the exponent. Thus, A"^ = A X A ; A^ 
= A X A X A, which is called the cube, or third power, of A. 

V This is called the square root sign: it signifies that the 
square root of the quantity covered by it is to be taken. 

V If preceded by a small secondary figure, called the index, 
as in the marginal figure, it indicates that the cube root of the 
quantity covered by it shall be taken; and so on. 

A^. If an exponent be fractional, as in the marginal figure, 
it requires that the square root of the third power of the quan- 
tity covered shall be taken, the numerator indicating the 
power and the denominator the root. 

B. M. Berbch-TnarJc : any fixed reference point for the level. 



ABBEE VIA TIONS. 

as outcropping ledge, water-table of building, or other perma- 
nent object. Usually a blunt conical seat for the rod, hewn 
on a buttressed tree-base, having a small nail sometimes driven 
flush in the top of it, and a blaze opposite, on which the eleva- 
tion is marked with kiel. 

T. P. Turning-point : usually marked O in the field-book. 

P. I. Point of intersection : as of tangents, which are to be 
connected by a curve. 

A. D. Apex distance: i.e., the distance from the P. I. toj 
the point where a curve merges in the tangent. 

P. C. Point of curve: the stake-mark at the beginning of] 
a curve. 

P. T. Point of tangent: the stake-mark at the end of a 
curve. 

P. C. C. Point of compound curvature : the stake-mark 
where a curve merges in another of different curvature, turn- 
ing in the same direction. 

P. R. C. Point of reverse curvature : the stake-mark where 
-a curve merges in another turning in the opposite direction. 

B. S. Backsight, in transit work ; or the reading of the rod 
to ascertain the instrument height in levelling. 

F. S. Foresight, in transit work; or the reading of the rod 
to ascertain elevations in levelling. 

H. I. Height of instrument : elevation of the level above 
the datum or zero plane. 

H. W. High water, 

L. W. Low water. 



V 



LOGAEITHMS. 
I.-II. 



i 



LOGAEITHMS. 



DEFINITIONS AND PRINCIPLES. 

1. The logaritlim of a number is the exponent of the powei 
to which it is necessary to raise a fixed number to produce the 
given number; that is to say, it represents the number of times 
a fixed number must be multiplied by itself in order to produce 
any given number. 

The fixed number is called the base of the system. In the 
common system, this base is 10. 

It follows from the above, that the logarithm of any power 
of 10 is equal to the exponent of that power. If, therefore, 
a number is an exact power of 10, its logarithm is a whole 
number. 

If a number is not an exact power of 10, its logarithm will 
not be a whole number, but will be made up of an entire part 
plus a fractional part, which is generally expressed decimally. 
The entire part of the logarithm is called the characteristic ; 
the decimal part is called the mantissa. 

2. The characteristic of the logarithm of a whole number 
is positive, and numerically 1 less than the number of places 
of figures in the given number. 

Thus, if a number lies between 1 and 10, its logarithm lies 
between and 1; that is, it is equal to phi.s a decimal. If a 
number lies between 10 and 100, its logaiithm is equal to 1 
plus a decimal ; and so on. 

3. The characteristic of the logarithm of a decimal fraction 
is negative, and numerically 1 greater than the number of O's 
that immediately follow the decimal point. 

The characteristic alone, in this case, is negative, the man- 
3 



4 MANNER OF USING THE TABLES. 

tissa being always positive. This is indicated by writing the 
negative sign over the characteristic : thus, 2.380211 is equiva- 
lent to — 2 + .380211. (See last example, p. 8.) 

4. The characteristic of the logarithm of a mixed number 
is the same as that of its entire part. Thus the mixed number 
74.103 Ues between 10 and 100; hence its logarithm lies be- 
tween 1 and 2, as does the logarithm of 74. 

5. The logarithm of the product of two numbers is equal to 
the sum of the logarithms of the numbers. 

The logarithm of a quotient is equal to the logarithm of the 
dividend diminished by that of the divisor. 

The logarithm of any power of a number is equal to the loga- 
rithm of the number multiplied by the exponent of the power. 

The logarithm of any root of a number is equal to the loga- 
rithm of the number divided by the index of the root. 

6. The preceding principles enable us to abridge labor in 
arithmetical calculations, by using simple addition and sub- 
traction instead of multiplication and division. 



II. 

MANl^ER OF USING THE TABLES. 

TO FIND THE LOGARITHM OF ANY NUMBER. 

1. First find the characteristic by rule 2, 3, or 4, given 
above. 

2. Then, if the number be less than 100, look in column N 
of the table for 10 times or 100 times the amount of it; oppo- 
site this multiple, in column O, will be found the mantissa. 

Thus the logarithm of 6 is 0.778151; that of 84 is 1.924279. 

3. If the number lie between 100 and 10000, find the first 
three figures of it in column N; then pass along a horizontal 
line until you come to the column headed with the fourth 
figure of the number. At this place will be found the 
mantissa. 

Thus the logarithm of 7200 is 3.857332; that of 8536 is 
3.931254. 



MANNER OF USING THE TABLES. 5 

4. If the number be greater than 10000, place a decimal point 
after the fourth figure, thus converting the number into a 
mixed number. Find the mantissa of the entire part by the 
method last given. Then take from cohunn D the correspond- 
ing tabular difference, multiply this by the decimal part, and 
add the product to the mantissa just found. The principle 
employed is that the differences of numbers are proportional 
to the differences of their logarithms, when these differences 
are small. 

Thus the logarithm of 672887 is 5.82794-3; that of 43467 is 
4.638160. 

5. If the number be a decimal, drop the decimal point, thus 
reducing it to a whole number. Find the mantissa of the log- 
arithm of this number, and it will be the mantissa required. 

Thus the logarithm of .0327 is 2.514548; that of 378.024 is 
2.577520. 

TO FIND THE NUMBER CORRESPONDING TO A GIVEN 
LOGARITHM. 

6. The rule is the reverse of those just given. Look in the 
table for the mantissa of the given logarithm. If it cannot be 
found, take out the next less mantissa, and also the corre- 
sponding number, which set aside. Find the difference be- 
tween the mantissa taken out and that of the given logarithm ; 
annex as many O's as may be necessary, and divide this result 
by the corresponding number in column D. Annex the quo- 
tient to the number set aside, and then point off from the left 
hand a number of places of figures equal to the characteristic 
plus 1 ; the result will be the number required. If the char- 
acteristic is negative, the result will be a pure decimal, and 
the number of O's which immediately follow the decimal 
point will be one less than the number of units in the charac- 
teristic. 

Thus the number corresponding to the logarithm 5.233568 
is 171225.296; that corresponding to the logarithm 2.233568 is 
.0171225. 

MULTIPLICATION BY MEANS OF LOGARITHMS. 

7. Find the logarithms of the factors, and take their sum; 
then find the number corresponding to the resulting logarithin, 
and it will be the product required. 



MANIiER OF USING THE TABLES. 

Example. 
Find the continued product of 8.902, 597.16, and 0.031472S. 

Ojjeration. 
Log. 3.902 . . . 0.591287 
Log. 597.16 . . . 2.776091 
Log. 0.0314728 . 2.497936 



1.865314 = log. 73.3354, the product. 

Here the 2 cancels the -j- 2, and the 1 carried from the deci« 
mal part is set down. 



DIVISION BY MEANS OF LOGARITHMS. 

8. Find the logarithms of the dividend and the divisor, and 
subtract the latter from the former; then find the number 
corresponding to the resulting logarithm, and it will be the 
quotient required. 

Example 1. 

Divide 24163 by 4567. 

Operation. 
Log. 24163 . . . 4.383151 
Log. 4567 . . . 3.659631 



0.723520 = log. 5.29078, the quotient. 

Example 2. 
Divide 0.7438 by 12.9476. 

Operation, 
Log. 0.7438. . . 1.871456 
Log. 12.9476 . . . 1.112189 

2.759267 = log. 0.057447, the quotient. 

Here 1 taken from ^ gives 2 for a result. The subtraction, as 
in this case, is always to be performed in the algebraic way. 

9. The operation of division, particularly when combined 
with that of multiplication, can often be simplified by using 
the principle of the arithmetical complement. 

The arithmetical comxAement of a logarithm (written a. c.) 



MANNER OF USING THE TABLES. 7 

is the result obtained by subtracting it from 10: it may be 
written out by commencing at the left hand, and subtracting 
each figure from 9 until the last significant figure is reached, 
which must be taken from 10. Thus 8.130456 is the arithmet- 
ical complement of 1.869544. 

To divide one number by another by means of the arith- 
metical complement, find the logarithm of the dividend and 
the arithmetical complement of the logarithm of the divisor; 
add them together, and diminish the sum by 10; the number 
corresponding to the resulting logarithm will be the quotient 
required. 

Example. 

Multiply 358884 by 5672, and divide the product by 89721. 

Operation. 

Log. 358884 . . . 5.554954 

Log. 5672 . . . 3.753736 

(a.c.)Log. 89721 . . . 5.047106 



4.355796 = log. 22688, the result. 
The operation of subtracting 10 is performed mentally. 

TO RAISE A NUMBER TO ANY POWDER BY MEANS OF LOGA- 
RITHMS. 

10. Find the logarithm of the number, and multiply it by 
the exponent of the power ; then find the number correspond- 
ing to the resulting logarithm, and it will be the power 
required. 

Example, 

Find the 5th power of 9. 

Operation. 

Log. 9 0.954243 

5 



4.771215 = log. 59049, the power. 



TO EXTRACT ROOTS BY MEANS OP LOGARITHMS. 

11. Find the logarithm of the number, and divide it by the 
Index of the root ; then find the number corresponding to the 
resulting logarithm, and it will be the root required. 



8 MANNER OF USING THE TABLES. 

Example. 
Find the cube root of 4,096. 

Operation, 
Log. 4,096, 3.612360; one-third of this is 1.204120, to which 
the corresponding number is 16, which is the root sought. 

12. When the characteristic is negative, and not divisible by 
the index, add to it the smallest negative number that will 
make it divisible, and then prefix the same number, with a 
plus sign, to the mantissa. 

Example. 
Find the 4th root of .00000081. The logarithm of this num- 
ber is 7.908485, which is equal to 8 -f 1.908485, and one-fourth 
of this is 2.477121 ; the number corresponding to this logarithm 
is .03: hence .03 is the root required. 

13. Five-figure logarithms are sufficiently accurate for ordi- 
nary railroad field-work. The tables in this book may there- 
fore, as a rule, be used without interpolation. 



PLANE TEIGO]SOMETET. 
III.— VIII. 



PLAJSTE TEIGO]SrOMETET, 



III. 

DEFINITIONS. 



1. Plane Trigonometry treats of the solution of plane tri- 
angles. 

In every plane triangle there are six parts, — three sides and 
three angles. When three of these parts are given, one beinj; 
a side, the remaining parts may be found by computation! 
The operation of finding the unknown parts is called the solu- 
tion of the triangle. 

10 



PLANE TRIGONOMETRY— DEFINITIONS. 



11 




2. A plane angle is measured by the arc of a circle included 
between its sides ; the centre of the circle being at the vertex, 
and its radius being 1. The circle, for convenience, is divided 
into 360 equal parts called degrees; 90 of these parts are 
included in a quadrant, which includes one-quarter of the 
circle, and is the measure of a right angle. Each degree is 
further divided into 60 equal parts called minutes, and each 
minute into 60 equal parts called seconds. Degrees, minutes, 
and seconds are de- 
noted by the symbols 
o, 'f ": thus the ex- 
pression 7° 22' 33'^ is 
read, 7 degrees, 22 
minutes, and 33 sec- 
onds. 

3. The complement 
of an angle is the dif- 
ference be;:ween that 
angle and a right 
angle. 

4. The supplement of an angle is the difference between 
that angle and two right angles. 

5. Instead of employing the arcs themselves, certain func- 
tions of the arcs are usually employed, as explained below. A 
function of a quantity is something which depends upon that 
quantity for its value. 

The sine of an angle is the distance from one extremity of 
the arc enclosing it, to the diameter, through the other extrem- 
ity. Thus P M is the sine of the angle M O A. 

The cosine of an angle is the sine of the complement of the 
angle. Thus N M = O P is the cosine of the angle M O A. 

The tangent of an angle is a right line which touches the 
enclosing arc at one extremity, and is limited by a right line 
drawn from the centre of the circle through the other extrem- 
ity : the sloping line which thus limits the tangent is called the 
secant of the angle. A T is the tangent and O T the secant of 
the angle MO A. 

The versed sine of an angle is that part of the diameter AP 
which is intercepted between the foot of the sine and the ex- 
tremity of the enclosing arc. 

The cotangent of an angle is the tangent of the complement 
of that angle; the co-versed sine and cosecant are similarly 
defined. Thus BT^ BN, and OT' are respectively the CQ' 
tangent, co-versed sine, and cosecant of the angle MO A. 



12 NATURAL SINES, ETC. 

These terms are in practice isdicated by obvious contractions; 
as, sin. A for the sine of A, cos. A for the cosine of A, &c. 

6. The above definitions have been made with reference to a 
radius of 1, Any function of an arc whose radius is R is 
equal to the corresponding function of an arc whose radius is 
1, multiplied by the radius R. So also any function of an arc 
whose radius is 1 is equal to the corresponding function of an 
arc whose radius is R, divided by that radius. 

7. Arcs greater than half a circle are not treated in this 
book, for the reasons that they seldom occur in field practice, 
and that any young engineer, conversant with the methods 
herein given, will, it is presumed, have no difficulty in band- 
ling such exceptional problems. Those curious in trigo- 
nometrical research should consult special treatises on the 
subject. 



IV. 

NATURAL SINES. ETC. 



1. Natural sines, cosines, tangents, or cotangents are those 
which are referred to a radius of 1. They may be used for all 
the purposes of trigonometrical computation ; but it is found 
more convenient, in many cases, to employ a table of logarith- 
mic functions. 



V. 

LOGARITHMIC SINES, ETC. 

1. Logarithmic functions are the logarithms of the natural 
functions treated in foregoing Art. IV., the characteristics 
throughout being transformed by the algebraic addition of 10, 
which has the effect of referring the tabular functions to a 
radius of 10, 000,000, 000 instead of unity, acid of thus, it |s sup- 



LOGARITHMIC SINES, ETC. 13 

posed, simplifying computation. For the intelligent use of 
these tables, however, it should be remembered that the char- 
acteristic 9 indicates a negative characteristic — 1, being one 
less than 10; a characteristic 7 indicates a negative character- 
istic — 3, and so on. Hence, in figuring with these logarithms, 
as many tens must be subtracted from the final result as have 
been added in the operation. This rule is illustrated in the 
following examples. 

TO FIND THE LOGARITHMIC FUNCTIONS OF AN ARC WHICH 
IS EXPRESSED IN DEGREES AND MINUTES. 

2. If the arc is less than 45°, look for the degrees at the top 
of the page, and for the minutes in the left-hand column; 
then follow the corresponding horizontal line till you come to 
the column designated at the top by sine, cosine, tang., or 
cotang., as the case may be; the number there found is the 
logarithm sought. 

Thus, log. sin. 19° 55^ . . . 9.582312 
log. tang. 19° 55' . . . . 9.55909T 

3. If the angle is greater than 45°, look for the degrees at 
the bottom of the page, and for the minutes in the right-hand 
column ; then follow the corresponding line towards the left, 
till you come to the column designated at the bottom by sine, 
cosine, tang, or cotang, as the case may be ; the number there 
found is the logarithm sought. 

Thus, log. cos. 52° 18' ... . 9.786416 
log. tan. 52° 18' ... . 10.111884 

4. If the arc is expressed in degrees, minutes, and seconds, 
proceed as before with the degrees and minutes ; then multiply 
the corresponding number taken from column D by the num- 
ber of seconds, and add the product to the preceding result, 
for the sine or tangent, and subtract it therefrom for tke cosine 
or cotangent. 

IJxample. 
Find the logarithmic sine of 40° 26' 28", 



14 LOGARITHMIC SINES, ETC, 

Operation. 

Log. sine 40° 26' 9.811952 

Tabular dife. 2.47 
No. of seconds,, 28 

Product . 69.16 to be added . 69 



' 



Log. sine 40° 26' 28'' 9.812021 

5. If the arc is greater than 90°, find the required function 
of its supplement. Thus the logarithmic tangent of 118° 18 
25", is equivalent to that of its supplement, or 61° 41' 35", 
and is 10.268732. Also the logarithmic cosine of 95° 18' 24" 
is 8.966078, and the log. cot. of 125" 23' 50'^ is 9.851619. 

TO FIND THE ARC CORRESPONDING TO ANY LOGARITHMIC 
FUNCTION. 

6. This is done by a reverse process. Look in the proper 
column of the table for the given logarithm; if it is found 
there, the degrees are to be taken from the top or bottom, and 
the minutes from the left or right hand column, as the case 
may be. If the given logarithm is not found in the table, find 
the next less logarithm, take from the table the corresponding 
degrees and minutes, and set them aside. Subtract the loga- 
rithm found in the table from the given logarithm, and divide 
the remainder by the corresponding tabular difference. The 
quotient vrill be seconds, which must be added to the degrees 
and minutes set aside, in the case of a sine or tangent, and 
subtracted in the case of a cosine or cotangent. 

Example. 
Find the arc corresponding to log. sin. 9.422248. 

Operation. 
Given logarithm . . . 9.422248 
Next less in table . . . 9.421857 . . . .15° 19' 






Tabular diff. . . 7.68) 391(51/' to be added. 
Hence the required arc is 15° 19' 51". 

7. By analogous process, the arc corresponding to log. cos. 
9.427485 will be found to be 74° 28' 43". 



GENERAL PROPOSITIONS. M 



VI. 

GENERAL PROPOSITIONS. 

1. In any right-angled triangle the hypothenuse is to one ot 
the legs as the radius to the sine of the angle opposite to that 
leg. 

And one of the legs is to the other as the radius to the tan- 
gent of the angle opposite to the latter. 

2. In any plane triangle, as one of the sides is to another, so 
is the sine of the angle opposite to the former to the sine of 
the angle opposite to the latter. 

3. In any plane triangle, as the sum of the sides about the 
vertical angle is to their difference, so is the tangent of half 
the sum of the angles at the base to the tangent of half their 
difference. 

4. In any plane triangle, as the cosine of half the difference 
of the angles at the base is to the cosine of half their sum, so 
is the sum of the sides about the vertical angle to the third 
side, or base. 

Also, as the sine of half the difference of the angles at the 
base is to the sine of half their sum, so is the difference of the 
sides about the vertical angle to the third side, or base. 

5. In any plane triangle, as the base is to the sum of the 
other two sides, so is the difference of those sides to the 
difference of the segments of the base made by a perpendicular 
let fall from the vertical angle. 

6. In any plane triangle, as twice the rectangle under any 
two sides is to the difference of the sum of the squares of 
those two sides and the square of the base, so is the radius to 
the cosine of the angle contained by the two sides. 



16 



SOLUTION OF PLANE TRJANGLSS. 



VII. 

SOLUTION OF PLANE TRIANGLES. 

1. It is usually, though not always, best to work the propor- 
tions in trigonometry by means of logarithms, taking the 
logarithm of the first term from the sum of the logarithms of 
the second and third terms, to obtain the logarithm of the 
fourth term; or adding the arithmetical complement of the 
logarithm of the first term to the logarithms of the other two, 
to obtain that of the fourth. 

2. There are three distinct cases in which separate rules are 
required. 

CASE I. 

3. When a side and an angle are two of the given parts, the 
solution may be effected by proposition 2 of the preceding 
section. 

If a side be required, say, — 

As the sine of the given angle is to its opposite side, 

So is the sine of either of the other angles to its opposite side. 

4. If an angle be required, say, — 

As one of the given sides is to the sine of its opposite angle, 
So is the other given side to the sine of its opposite angle. 
The third angle becomes known by taking the sum of the 
two former from 180°. 

Example 1. 
Given angle A = 24° 26'; angle 
B = 36° 43'; side 6 = 137.6: to find 
side a. 




As sin. B . 
Is to sin, A 
Sois& . . 



To a, 95.2 



log. 9.776598 
log. 9.616616 
log. 2.138618 

11.755234, sum of 2d and 3d terms, 

log. 1.978636 less 1st term. 



SOLUTION OF PLANE TRIANGLES, H 

Example 2. 
Given, sides a and 6, as above, and angle A ; to find angle B. 

As side a (a. c.) log. 8.021364 

Is to sin. A log. 9.616616 

So is side b log. 2.138618 



To sin. B = 36° 43' log. 9.776598 sum. 



CASE IT. 

5. When two sides and the included angle are given, the 
solution may be effected by means of propositions 3 and 4. 
Thus, take the given angle from 180°; the remainder will be 
the sum of the other two angles. 

Then, by proposition 3, — 

As the sum of the given sides is to their difference, 

So is the tangent of half the sum of the remaining angles 
to the tangent of half their difference. 

Half the sum of the remaining angles added to half their 
difference will give the larger of them, and half their sum 
diminished by half their difference will give the lesser of them. 

The solution may be completed either by proposition 4, or 
by proposition 2, as in Case I. 

Example. 
Given side a =95.2, side 6^137.6, and the included angle 
c = 118° 51'; to find the remaining angles. Here 180.00 — 
118° 51' =61° 09', the sum of the remaining angles. 

As sum of given sides, 232.8 log. 2.366983 

Is to their difference, 42.4 log. 1.627366 

So is tang. \ sum of rem. angles, 30° 34^' . log. 9.771447 

To fane/. I their difference = 6° 081^ . . . log. 9.031830 

Adding half the difference to half the sum, 30° 34i' + 6c 
08^' = .36° 43', = the larger angle, B. Deducting half the 
difference from half the sum =24° 26' = the smaller angle, A. 

This case is susceptible of solution also by means of propo- 
sition 6. 



18 RIGHT-ANGLED PLANE TRIANGLES. 



CASE Til. 

6. When the three sides of a plane triangle are given, to 
find the angles. 

First Method. 

Assume the longest of the three sides as base; then say, 
conformably with proposition 5, — 

As the base is to the sum of the two other sides, 

So is the difference of those sides to the difference of the 
segments of the base. 

Half the base added to half the said difference gives the 
greater segment, and diminished by it gives the less ; thus, by 
means of the perpendicular from the vertical angle, the original 
triangle is divided into tM'O, each of which falls under the first 
case. Or they may be solved by the simpler methods applica- 
ble to right-angled triangles. 

Second Method. 

7. Find any one of the angles by means of proposition 6, and 
the remaining angles either by a repetition of the same rule, 
or by the relation of sides to the sines of their opposite angles. 



VIII. 
EIGHT-AXGLED PLANE TRIANGLES. 

1. Right angles may be solved by the rules applicable to all 
plane triangles; and it will be found, since a right angle is 
always one of the data, that the rule usually becomes simplified 
in its application. 

2. When two of the sides are given, the third may be found 
by means of the rule that the square of the hypothenuse is 
equal to the sum of the squares of the remaining sides. 

3. Another method for solving right-angled triangles is as 
follows: — 

To find a side. Call any one of the sides radius, and write 
upon it the word " radius." Observe whether the other sides 




RIGHT-ANGLED PLANE TRIANGLES. 19 

become sines, tangents, cosines, or the like, and write upon 
them tlie proper designations accordingly. Then say. 
As the name of the given side is to the given side. 
So is the name of the required side to the required side. 

4. To find an angle. Assume one side to be radius, and 
mark the remaining sides as before. Then say, 

As the side made radius is to radius. 
So is the other given side to the name 
of that side; 
Which determines the opposite angle. 

5. Applying this method to the right- 
angled triangle ABC, and calling the 
hypothenuse a radius, we shall have, 

c = a sin. C -f- R; hence sin. C = Re -f- a. 
b = a COS. C -i- R; hence cos. C = Rb -4- a. 

Then, assuming the side b to be radius, we shall have, 

c = b tang. C -4- R; hence tang. C= Re -f- b. 

If radius be called 1, the natural sines and cosines will be 
used in the application of these formulas ; they are often more 
convenient than logarithms in railroad practice, especially 
when the numbers which measure the sides of the triangle are 
either less than 12, or are resolvable into factors less than 12. 

6. The simpler relations between these natural functions are 
as follows : 

sm.^ + cos.^ = 1; tang. = — ; cotang. = -r- = 
' COS. sm. 



11., . , . 

— : cosec. — -7— ; versme = \ — cos, ; GO-versine= 1 — sin. 
COS. sm. 



ADJUSTMENT AND USB 

OF 

INSTRUMENTS, 
IX.-XV. 



ADJUSTMENT AND USE 

OF 

INSTRUMENTS. 



IX. 

GENERAL REMARKS ON ADJUSTMENTS. 

1. Care should be taken in all instrumental adjustments, 
where screws work in pairs, to loosen one before tightening its 
opposite. 

2. Remember that the eye-piece inverts the image of the 
cross-hairs, and that consequently any movement of it, by 
means of the small capstan head screws on the outside of the 
telescope-barrel, should be in the direction which would seem 
to increase the error requiring correction. 

3. Before beginning the adjustments, screw the object-glass 
close home, and make a pin-scratch across its rim and the end 
of the tube, by which to mark its proper place ; draw out the 
eye-piece until the cross-hairs are exactly in focus ; that is to 
say, until no movement of the eye shall appear to displace 
them, and bring the object to be observed clearly into view. 

4. Never permit the glasses to be rubbed with a gritty fabric. 
To remove the dust from them, use a soft, clean handkerchief, 
and change often the part applied. 



24 THE LEVEL, 

X. 

THE LEVEL. 

ro BRING THE INTERSECTION OF THE CROSS-HAIRS INTO 
THE OPTICAL AXIS OF THE TELESCOPE. 

1. Set the instrument firmly, cast loose the wyes, and, by 
>evelHng and tangent screws, bring either of the cross-hairs to 
coincide with a well-defined object, distant from 400 to 000 
feet, or as much farther as distinct vision can be had free from 
heat ripple. Gently rotate the telescope half-way around in 
the wyes. If the cross-hair selected for treatment then fails 
to coincide with the object, reduce the error one-half by means 
of the small capstan head screws at right angles to it on the 
telescope-barrel. Bring the spider-line again to coincide with 
the object by means of tbe levelling and tangent screws, and, 
if necessary, repeat the operation. Proceed in the same man- 
ner with the other cross-hair. If the error is large, bring both 
nearly right before undertaking their final adjustment. 

2. Having thus adjusted the line of collimation upon a dis- 
tant point, requiring the object-tube to be drawn well in, select 
a point close by, which shall require it to be thrust out almost 
to its limit. If any error appears, correct half of it with the 
small screws provided for the purpose, a little forward of the 
diaphragm, and usually protected by a movable sleeve on 
the outside; correct the other half with the levelling-screws. 
After completing this adjustment, test the former one on a 
distant object, and. if necessary, repeat the operations. 

3. In the transit, the small guide-ring screws used for this 
adjustment are covered by the bulb of the cross-bar in which 
the telescope is fixed, and are therefore inaccessible. The 
adjustment, however, is one not liable to become deranged in 
either instrument, and, in the transit, is of comparatively 
small importance. 

4. The young practitioner should bear in mind that the 
intersection of the cross-hairs may coincide with the optical 
axis of the telescope, and yet be out of centre as regards the 
field of view. Such' eccentricity does not affect the working 
accuracy of the instrument, which depends upon the position 



TUE LEVEL. 25 

Df the object-piece solely. It may be removed by manipulation 
of the small screws securing the inner end of the eye-piece. 

TO BEING THE LEVEL BUBBLE PARALLEL WITH THE TELE- 
SCOPE AXIS. 

5. Clamp the instrument over either pair of levelling screws, 
and bring the bubble to the middle of its tube. Turn the tele- 
scope slightly on its bearings, so that the bubble-case shaii 
project a little on one side or the other. If the bubble sli]is, 
coi-rect half its movement by means of the small lateral capstan 
head screws at one end of the case. Return the telescope tc 
its first position, level up again, and repeat the operation until 
the erroneous movement ceases. This adjustment brings the 
telescope and level into the same vertical plane. 

G. Next, the bubble being at the middle of its tube, carefully 
lift the telescope out of the wyes, turn it end for end, and 
replace it. If the bubble settles away from the middle, bring 
it half-way back by means of the capstan-heads, working up 
and down at one end of the case. Again middle it with the 
levelling screws, and repeat the opei-ation until the error is 
corrected. 

rO ADJUST THE WYES ; OE, IN OTHER WORDS, TO BRING THE 
TELESCOPE INTO A POSITION AT RIGUT ANGLES TO THE 
VEETICAL AXIS OF THE INSTRUMENT. 

7. Close the wyes. Unclamp. Set the telescope directly 
over two of the levelling screws, and with them bring the 
bubble to the middle of the tube. Then rotate the telescoi^o 
horizontally, until it stands over the same pair of screws, 
changed end for end. If the bubble errs, correct one-half of 
the deviation with the capstan head nuts at the end of the 
bar, and one-half with the levelling screws. Place the tele- 
scope over the other pair of levelling screws, liepeat the 
operation there; and continue the corrections, over one and 
the other pair of levelling screws alternately, until the bubbk' 
stands without varying during an entire revolution of Hk- 
instrument upon its vertical axis. 

8. The capstan head nuts on the cross-bar should be moved 
by gradual stress, not by pounding. They are a I'ude deviec 
With so short a leverage as the length of the common adjusi- 
ing-pin suppUes, it is almost impossible to give them a smooth. 



26 LEVELLING. 

manageable motion. They reproach the instrument-maker's 
art as unchecked hydrophobia and cancer do that of medicine, 
or mercenary villany that of law, and should be supplanted by 
better practice. 

9. Having thus completed the principal adjustments in their 
proper order, bring the telescope and its l)ubble-case as nearly 
vertical in the wye bciunigs as hand and eye can make them, 
and by reference to a plumb-line, or the corner of a well-built 
house, see if the vertical hair is out of true. If so, slightly 
loosen two opposite screws of the diaphragm, and correct the 
error by turning it. Try again the adjustment of the line of 
collimation before pinning up the wyes. 



XI. 

LEVELLING. 



1. Suppose O the starting-point; 1, 2, 3, &c., the stakes of 
survey; and A the initial bench-mark. Wherever convenient 
the elevation of A above mean tide should be ascertained. 
It is to be regretted that this was not done from the outset. 




under statute provisions requiring maps and profiles also to 
be filed at the several State capitals. In that case, not only 
would much after labor and expense by way of duplicate sur- 
veys have been spared, but the older Commonwealths would 
now have in hand materials for the preparation of physio- 
graphical maps, the value of which to science, to the engineer, 
and to the economical geologist, it were hard to measure. 



LEVELLING. 



27 



2. For the purposes of a railroad-survey, however, such 
determination is not needful. Any elevation may be assumed 
for A, taking care only that it be large enough to avoid the 
possibility of having minus levels, which would be inconven- 
ient. Zero of the datum should be below the lowest probable 
ground on the contemplated line. 

3. Let the elevation of the initial bench-mark, A, in the 
figure, be taken at -f-200. Set the level at B, and suppose the 
rod on the BM to read 2.22. The " instrument height " then 
is 202.22. If the rod at sta. O reads 8.4, the elevation at that 
point is 202.22 — 8.4 = 193.8. The rod reading 1.9 at sta. 1, 
the elevation there is 202.2 — 1.9 = 200.3. If desirable to turn 
at sta. 2, drive a pin nearly to the ground at that stake ; sup- 
pose the rod on it to read 0.81. The elevation then is 202.22 — 
0.81 = 201.41. Now move the instrument to C, and, sighting 
back to sta. 2, let the rod standing on the pin read 2.64. This 
makes the new " instrument heiglu" at C = 201.41, the height 
of sta. 2, -f 2.64 == 204.05, and the elevations at 3, 4, 5, or 
other points observed from C are found by deducting the 
readings at those points from the ascertained instrument height 
at the new point of observation. 

" 4. It thus appears how simple is the rule of levelling, 
namely: Find the "instrument height" by adding the "back- 
sight" to the elevation of the point upon which the rod stands 
for that purpose : from the "instrument height" thus found 
deduct the " foresights," severally, in order to find the eleva- 
tions of the points at which such foresights are taken. 

5. The foregoing example woidd appear in the field-book as 
follows : — 



Sta. 


B. S. 


Inst. 


F.S. 


Eleva. 


Remarks. 


BM 








200.00 


B M on W. Oak. 




2.22 


202.22 






40 ft. N. of Sta. O. 





.. 


.. 


8.4 


193.8 




1 






1.9 


200.3 




2 


2!64 


204 '.05 


0.81 


201.41 




3 


.. 




3.7 


200.3 




4 


.. 




3.2 


200.8 




5 






10.36 


193.69 


1 



G. In levelling where great exactness is necessary, the rod at 
t-uruing-points should be read to thousandths, and the reading 
<ihecked by the leveller. Before taking it down, after clamp- 



28 LEVELLING. 

iiig tlie target fast, it shoitld be swayed slowly to and fro in the 
direction of the instrument to make snre of getting the full 
height. In foul weatlier the rodnian should take care that 
the foot of the rod does not ball up with mud or snow. The 
leveller should have his cross-hairs free from parallax, the tar- 
get in focus, and see his bubble true at the moment of obser- 
vation. He should also set the instrument about half-way 
between turning-points when practicable, balancing lar<Tely 
unequal sights l)y subsequent ones similarly unequal in the 
opposite direction; and his turning-points, even on favorable 
ground, ought not to be more than 000 or 800 feet asunder, 

7. On ordinary railroad field vvoi-k such nicety as is implied 
in most of these rules is not required. To read to the nearest, 
tenth is suflicient, especially where the progress of the parly 
depends in a good degree on the level; as, for example, in run- 
ning grade lines on preliminary survey. The location levels 
are usually carried along more carefully; but even then the 
writer's practice has been to turn to hundredths only. 

8. The Philadelphia Rod is the best for our se'rvice. The 
sliding lialves are unconnected except by brass sleeves or 
clips, which guide them, and are therefore not liable to bind in 
wet weather. They are made by William J. Young's ^ons, 
who some years ago, at the writer's suggestion, supplied what 
seemed to be their only defect by adopting rivets for fastening 
the clips instead of wood screws; the screws had a tendency lo 
work loose in the field, and cause the parts to chafe or jam. 
These rods are cleaiiy figured, so as to be legible at a distance 
of several hundred feet; the leveller is thus enabled to take 
intermediate elevations rapidly, and, when necessary, to do 
his work with the aid of an unlettered rodmau. 



9. cokliection fon the earth 's curvatuke and refkac- 

t;ox. 

The correction for a 100-feet "station" is .000215; for one 
mile, 0.(). It is to oe added to the calculated elevation of tlie 
point observed, or to be deducted from the "rod" before 
calculatuig the elevation, in the case of a long unbalanced 
sight. It varies as the square of the distance. Cahing the 
requiied correction A, for any given distance D, then A = 
.000215 X DMf U is hi ^' stations," and A = 0.6 X D^ if D 
is in miles. Thus the correction for 10 stations would be 



LEVELLING. 29 

• 

,0215; for 50 stations, 0.5375; for 10 miles, 60 feet, and a spire 
or treetop apparently level with the instrument at that dis- 
tance would really be (30 feet above it. Ti'ansposing the equa- 
tion we have D = V^A-^0.0. In this form it is applicable to 
the dtttermination of distances at sea. The Peak of Teneriffe, 
for example, 16,000 feet high, should be just visible from the 
sea- level at a distance = y' 16000 ^0.6== say 163 miles. 

10. TO FIND DIFFEREN<:ES in elevation liY MEANS OF THE 
CAUOMETEn. 

Call the required difference D; the barometrical reading 
at the lower stand, L; that at the upper stand, U. 

Then, D = (L — U) ^ (L + U) X 55000. 

Examine. 
L = 26.64; U = 20.82. 

Then, L — U= 5.82 .... log. 0.764923 
L + U = 47.46 .... log. 1.676328 

0.1226 Diff. —1.088595 

And 0.1226 X 55000 = 6743, the required difference of elevation 
in feet. 

11. A closer approximation is thought to be attainable by 
using a thermometer in connection with the mercurial barome- 
ter. In that case, having found the difference as above, add 
^1q of the result for each degree by which the mean tempera- 
ture of the air at the two stands exceeds 55°; subtract the like 
proportion if the mean temperature be below 55°. When the 
upper thermometer reads highest, for "subtract" say "add," 
and vice versa in the foregoing rule. 

12. The naked formula, however, will usually be sufficient 
for the engineer. He can prescribe gradients by it for surveys, 
which shall develop the ground to be occupied, and can decide 
between summits well differenced in height. If not so differ- 
enced, questions of detour, of approaches, and the like, will 
contribute to determine the expediency of making an instru- 
mental examination. 

13. HEIGHTS BY THE THERMOMETER. 

T=: the difference, in degrees Fahrenheit, between 212°, the 
t;imi3erature of boiling water at the sea level, and that at the 
J) I ace of observation. 



30 SETTING SLOPE STAKES. 

H = the height of place of observation above or below thfl 
sea ill feet. 

11 = 513 T 4- T2. 

Example. 
T = 212° — 208° = 4°. 
H = (513 X 4) + 4'^ = 2068 feet. 



XII. 

SETTING SLOrE STAKES. 

1. Like swallowing, this is more easily done than described. 
To no detail of field service does the proverb more fitly apply, 
that "work makes the workman." 

2. The problem is, to find where a formation slope of given 
Inclination, beginning at the side of the road-bed, must needs 
intersect the ground surface. Formation slopes are usually 
stated in parts horizontal to one part vertical. Thus a slope 
of 45° is " 1 to 1." A slope of " 2 to 1 " has a horizontal reach 
of two feet to each foot vertical. The carriages of a stairway 
with twelve-inch treads and eight-inch risers would have a 
slope of "Hto 1." 

3. To fix the point where any proposed formation slope must 
meet the surface on level ground, is quite simple; the distance 
from tlie centre line being obviously made up of half the width 
of road-bed added to the horizontal distance due from the 
slope, to the depth of cut or lieight of fill. Thus, with 20 feet 
road-bed, 9 feet cut, and slope of 1^ to 1, the distance out 
would be 10 -[- 9 + 4^ = 23^ feet, as shown in the annexed 
diagram. , 



K-^^ 

# 



4. On slant or broken ground, the solution is more difficult: 
recourse must then be had to the level, with a rodmau, a tape- 
man, and, for good speed, an axeman to assist. 



SETTING SLOPE STAKES. 



31 



Example No. 1. 
5. Let the accompanying figure represent the cross-section at 
\ any given point of a proposed excavation ; road-bed 20 feet wide, 
j cutting at centre stal^e 12 feet, and formation slopes 1 to 1. 




6. The first step is to set the level, as at A, commanding, let 
us suppose, the lower slope, and to ascertain its height above 
grade at the proposed section. This is usually done by refer- 
ence to the nearest bench, and pegging from stake to stake as 
the work progresses. Unless the ground is very steep, and the 
slope-stakes largely different in elevation, labor will be saved 
and likelihood of error reduced by levelling over the centre 
line beforehand, as a separate job, and marking on centre 
stakes the cuts, fills, and grade points, that is to say, the points 
where excavation passes into embankment. The rods should 
be taken carefully at the stakes, and the latter marked on 
their backs to the nearest tenth, as "grade," " C 12," signify- 
ing cut n feet, or "F 6.2," signifying fill 6.2 feet, for ex- 
ample. This being done, each centre stake serves as a bench- 
mark for slope staking at that section, and each cross section 
can be staked out independently. 

7. Instrument height, in the example treated, being by either 
method fixed at 15.5 above grade, the next step is a guess how 
far out from the centre stake the formation slope would proba- 
bly meet the ground surface. The closeness of the guess will 
correspond to the experience and natural skill of the leveller: 
the young engineer should not be discouraged if he misses tj « 
mark rather widely in liis early trials. 



32 HETTINQ SLOPE STAKES. 

8. It is true, that, on a uniform declivity, he might aid con- 
jecture by taking a rod distant half the width of road-bed, or 
10 feet, from the centre stake, ascertain thus the slope of the 
::round surface as well as the cutting at that point; and with 
iliese data, knowing also the formation slope, approximate 
I he point sought by solving the terminal triangle of the pro- 
posed section, indicated by dotted lines in the figure. But, in 
practice, he will find it the quicker and better way to approxi- 
mate the point by vividly imagining the underground forma- 
tion lines; or by vividly imagining a level section, the upper 
surface of which shall coincide with his instrument height, 
15.5 feet above grade. This gives him a point in the air, 
10 + 15.5 = 25.5 feet out from the centre stake, level with the 
instrument, as the limit of the imaginary section; and from 
that point he can pretty well judge where a line corresponding 
to the formation slope must meet the ground. 

\). Suppose him, by either method, or even by random guess. 
- to think that 10 feet for half the road-bed, and 10 more for 
the slope, looks about right. The formation slope being 1 to 
1, this implies a cutting of 10 feet at the side stake, and a rod, 
therefore, of 15.5 — 10.0 = 5.5 feet. Taking a rod accordingly, 
20 feet out, measured horizontally from the centre stake, he 
finds it to be 11.0 instead of 5.5, indicating that he has gone 
too far down hill. Let him now reason that the rod of 11.0 
corresponds to a cutting of 15.5 — 11.0 = 4.5 feet, and that a 
cutting of 4.5 feet corresponds to a distance out of 10 -|- 4.5 
= 14.5 feet. Try, then, a rod 14.5 feet out. It proves to be 
D.O, corresponding to a cutting of 15.5 — 9.0 = 6.5, instead of 
4.5 feet, and a distance out of 16.5 instead of 14.5 feet. Try, 
next, 16.5 feet out; the rod there, of 10.0 instead of 9.0, shows 
him again to be in error on the down-hill side of his object; 
but the lessening error shows also that he is approaching it, 
and that a few more like trials will reach it. 

10. Recurring to his first error with the 11.0 feet rod, he 
cannot fail to observe after a little practice, since the ground 
ascends thence toward the centre line, that the side stake 
must fall farther out than the point where his second trial w as 
made ; that its true position, in fact, divides the distance be- 
tween those points of observation into two parts which are to 
one another directly as the inclinations of the formation slope 
arid the ground surface. By degrees he will grow skilful in 
diviuing this true position, and, becoming meauwiiile quick Id 



SETTING SLOPE STAKES. 



3c 



observation, will place a slope stake on the second or third 
trial, without conscious effort of mind. 

11. Next, suppose the level at B, 25.5 feet above grade, com- 
raanding the upper slope. 

Note the change of ground 11 feet out, and take a rod there, 
recording the observation. The cutting at that point is 
25.5 — 9.5 = 16 feet, corresponding to a distance out for the 
side stake of 10 + 16 = 26 feet, if the ground wei-e level. A 
trial rod 26 feet out reads 7.8, corresponding to a cutting of 
25.5 — 7.8 = 17.7 feet, and a distance out for the side stake 
of 10 + 1*7." = 27.7 feet, showing that the point sought is still 
beyond. A repetition of such trials will finally fix it; but, as 
HI the case of the lower slope, practice will speedily lessen the 
number and abridge the labor of them. 

12. The foregoing section would be noted in the field book 
as follows : — 



Sta. Dis. 


Left. 


Centre 


Right. 


Area. 


C.Yds 


258 50 


4-5.8 
15.8 




+ 12.0 


+ 16.0 
11.0 


+ 18.0 
28.0 







Example No. 2. 
13. In the annexed figure, representing an embankment 14 
feet wide on top, with side slopes of 1| to l,the first thing to 
attract attention is that the instrument is 1 foot helow grade, 



-30,0 — 

--25i0- 
23.5- 



7^' 



s?-^?-- 



.^° 



and that, therefore, 1.0 is to be added to all rods, in order to 
find the height of embankment above the points at which rods 
are taken. 

14. Consider the down-hill side. The engineer, with the 
ground in view, and witli tlic height of embaukmeut Jvt tho 



34 



SETTING SLOPE STAKES. 



centre sta^e to aid him in forming an airy image of the pro- 
posed section, judges that the natural surface and the forma- 
tion slopes will meet 80 feet out. Of this distance, 7 feet are 
due to half the road-bed, and 23 feet to horizontal reach of the 
embankment slope. The slope being 1| to 1, or |, the hori- 
zontal reach of 23 feet corresponds to a vertical height of 
f of 23 = 15.3 feet; and, siiice the instrument is 1 foot be- 
low grade, to a rod at the supposed embankment base of 
153 — 1.0= 14.3 feet. But the rod at that point is only 11 
feet, to which, if 1 foot, the distance of instrument below 
grade, be added, the height of embankment would be 12 feet. 
He may then, as in the case of the upper slope in Example No, 
1, try a rod at the distance out corresponding to the 11 feet 
rod, or 12 feet embankment. This distance would be 7 + -- 
-(-6 = 25 feet, where, on trial, the rod proves to be 10 feet, 
instead of 11 feet, corresponding to an embankment height of 
10 -|- 1 = 11 feet, and to a distance out of 7 + 11 + 5 5 = 23.5 
feet. Approximating thus, by shorter and shorter steps, ne 
finally reaches the point sought. 

15. The process in fixing the upper slope stake is similar to 
that used in fixing the lower one in Example No. 1. The 
several steps are designated by small letters in the figure, and 
a detail of them is not thought necessary. 

16. This section would be noted in the field book as fol 
lows: — 



St A. 


Dis. 


Left. 


Centre 


Right. 


Area. 


C.Yds. 


140 


62 


— 9.4 

22.6 




— 6.3 




-3.2 
12.7 







Example No. 3. 

17. Here is a case, partly in rock excavation, slope |tol; 
partly in embankment, slope 1^ to 1; road-bed 17 feet wide, 9 
feet of which are on the right of tlie centre line and 8 feet on 
the left. 

IS. For the lower slope suppose the instrument height at A 
to be 6.5 feet above grade ; centre cutting 2.5 feet. Find first, 
with a 6.5 feet rod, the grade point, to left of oentre line, 
which proves to be 2.5 feet out. Xote it, and set a stake 
there marked "grade." Note also the change of ground 5.5 



SETTING SLOPE STAKES. 



35 



feet out and 10.0 — 6.5 = 3.5 feet beloAv grade. Then set the 
lower slope stake as in Example No. 2, observing that in this 




case the instrument is above grade, and that its height above 
grade is to be deducted from the rod at any point in order to 
obtain the height of grade above such point. 

19. Move the instrument to B, say 22.5 feet above grade. 
This elevation, if the cutting on that side be deemed to equal 
it, corresponds to a distance out of 9 feet for road-bed added to 
(22.5 -^ 4) for slope; total, 14.6 feet. The trial rod, however, 
at that distance, instead of reading 0, reads 6 feet, indicating a 
cut 22.5 — 6.0 = 16.5 feet deep, and a distance out correspond- 
ing thereto of 9.0 -\- (16.5 -i- 4) = 13.1 feet. Trying again at 
this distance out, the rod reads 7.6 instead of 6 feet, requiring 
a further movement towards the centre line of (7.6 — 6) -f- 4 
= 0.4 feet. Thus by approximations much more rapid than in 
the case of a flatter formation slope, the point is soon fixed. 

20. The field record of the above is as follows : — 



Sta. 


Dis. 


Left. j 


Centre 


Right. 


Area. 


C.Yus. 


328 


40 


t 

— 6.9 
18.3 


0.0 1 

■2.5 I 

-3.5 f 

5.5 J 


+ 2.5 




4-15.0 
12.8 







VERTICAL CURVES. 



XIII. 



VERTICAL CURVES. 



DIAGRAM. GIVING THE ORDINATES 
TERVALS OF -^-^ TO THE SPAN, 
BEING UNITY. 



OF A PARABOLA AT IN- 
THE MIDDLE ORDINATE 




1. Suppose gradients doscending right and left at an equal 
rate from the summit B, and lliat it is requii-ed to truncate the? 
summit with a vertical curve extending 150 feet each Avay. 

A circular arc consuming so small an angle may be treated 
as a parabola, in which the external secant B F is equal to the 
versed sine FT). Referring to the above diagram, ordinates 4 
and 8 will be seen to correspond to the ordinates between 




chord AC and the curve in this instance, which ordinates 
therefore will be equal to the middle ordinate FD multiplied 
by 0.89 and 0.55 respectively. Adding these multiples to the 
grade elevation at A, the elevations of the intermediate poiiitg 
selected will be ascertained. 



VERTICAL CURVES. 



Example 1. 
Elevation at A = + 94.0 ; A B = + 1 in 100; B C = — 1 in 
JOO; AD, DC, each = 150 feet or 1.5 stations of 100 feet each. 

Hence BD = 1.5; and FD =0.75 feet. 

Ordinate S = 0.75 X 0.55 = 0.41. 

Ordinal e 4 = 0.75 X 0.89 = 0.67. 
Elevation of grade at 8 — 8 = 94.0 + 0.41 = 94.41. 
Elevation of grade at 4 — 4 = 94.0 + 0.G7 = 94.67. 
Elevation of grade at D = 94.0 + 0.75 = 94.75. 

Example 2. 




Elevation at A = + 94.0. AB = + 1 in 100; B C = — 0.4 
in 100; AH, level; AD, DH, each.= 200 feet, or 2 stations, 
divided into 50 feet spaces, the points of division correspond- 
ing therefore to ordinates 3, 6, and 9 of the preceding diagram. 



CII 



1 X 2 — 0.4 X 2 = 2.0 — O.S = 1.2 feet. 



1.2 H- 



Ascent from A to C along chord A C = C H H- 8 

8 = 0.15 per 50 feet. 

BE = BD— iCH = 2 — 0.6 = 1.4. 
.-. FE = 1.4-f-2 = 0.7. 
Ordinales at 9 — 9 = 0.7 X 0.44 = 0.31. 
Ordinates at 6 — 6 = 0.7 X 0.75 = 0.52. 
Ordinates at 3 — 3 = 0.7 X 0.94 = 0.66. 
Mid-ordinate = 0.70. 

The elevations then along the chord AC, ascending at the 
i-ate of 0.15 per 50 feet, will be : — 

A 963 0369 C 

94.0 94.15 94.30 94.45 94.60 94.75 94.90 95.05 95.20 



ss 



VERTICAL CmV^ES. 



to wliicb add the ordinates just found: — 

0.0 0.31 0.52 0.6G 0.70 0.66 0.52 0.31 0.0 
Rud the grade elevations on the curve M'ill be: — 

94.0 94.46 94.82- 95.11 95.30 95.41 95.42 95.36 95.2 

Example 3. 
Elevation at A = + 94.0 ; A B = + 1 in 100 ; B C, A H, level. 
ID, B C, each 200 feet divided into 50-feet spaces, the points 




f{ division corresponding therefore to ordinates 3, 6, and 9 of 
,he ordinate diagram CH = 1X2 = 2 feet. 

Ascent from A to C along chord AC = CH-^8 = 0.25 per 
60 feet. 

BE = BD — ^CH = 1 foot. 
.-. FE = 1 -^2 = 0.5. 
Ordinates 9 — 9 = 0.5 X 0.44 = 0.22. 
Ordinates 6 — 6 = 0.5 X 0.75 = 0.37. 
Ordinates 3 — 3 = 0.5 X 0.94 = 0.47. 
Mid.ordinate = =0.50. 

The elevations then along the chord A C, ascending at the 
rate of 0.25 per 50 feet, will be: — 

A9 63 36 9C 

94.0 94.25 94.5 94.75 95.0 95.25 95.5 95.75 96.0 

to which add the ordinates just found: — 

0.0 0.22 0.37 0.47 0.5 0.47 0.37 0.22 0.0 

And the grade elevations on the curve will be : — 

94.0 94.47 94.87 95.22 95.5 95.72 95.87 95.97 96.0 



i 



VERTICAL CURVES. 



39 



Example 4. 
Elevation at A = + 94.0; AB =— 1 in 100; BC, AH, 
level; AD, BC, each 150 feet, divided into 50-feet spaces, the 
points of division corresponding therefore to ordinates 8 and 4 
of the initial diagram C H = 1 X 1.5 = 1.5. 




Descent from A to C along chord A C = C H ^ 6 = 0.25. 

^ B = D B — D E = 1.5 — 0.75 = 0.75 

.-. FE = 0.75 ^2 = 0.375 
Ordinates 8 — 8 = 0.375 X 55 = 0.21 
Ordinates 4 — 4 = 0.375 X 89 = 0.33 
Mid ordinate = 0.37 

The elevations then along the chord A C, descending at the 
rate of 0.25 per 50 feet, will be: — 



A 8 4 

94.0 93.75 93.5 



4 

93.25 93.0 



8 C 

92.75 92.5 



From which deduct the ordinates just found, 

0.0 0.21 0.33 0.37 0.33 0.21 0.0 
And the grade elevations on the curve will be : — 

94.0 93.54 93.17 92.88 92.67 92.54 92.5 

The figures are drawn much distorted, in order to make the 
illustration clear. 

2. With profile paper at hand, a convenient and quite suf- 
ficient determination of the grade elevations on a vertical 
curve may be made by drawing the gradients to a scale of 2 
feet to an inch vertical, and 50 feet to an inch horizontal. By 
means of the curve protractor (Art. XXV. 1) a suitable arc may 
then be fitted and struck in, and the elevations read ofL 



40 TEE TRANSIT, 



XIV. 

THE TRANSIT. 

1. Should the vernier and circle plates be out of parallel, — 
should one or the other be sprung, a defect shown by a slight 
rocking motion when the rims are pinched alternately on op- 
posite sides, — the instrument must be sent to the shop for 
repair. This is a common disease of transits in their old age: 
instrument-makers need to study its cause and cure. 



2. TO ADJUST THE LEVEL TUBES. 

Bring the bubbles to the middle of them by means of the 
levelling screws. Turn the top of the instrument horizontally 
iialf way round. If the bubbles then err, reduce the error one- 
half with the small adjusting screws attached to the tubes, 
and one-half with the levelling screws. Repeat until the ad- 
justment is perfect. 



3. TO ADJUST THE VERTICAL HAIR SO THAT IT SHALL RE- 
VOLVE IN A PLANE, AND MARK BACKSIGHT AND FORE- 
SIGHT POINTS IN THE SAME STRAIGHT LINE. 

Try, first, by reference to the corner of a well-built house, 
or to a plumb-line, whether the hair be truly vertical. If it is 
not, loosen the four small capstan head screws on the outside 
of the barrel slightly, and gently tap the topmost one right or 
left, until the adjustment is effected. 

4. Then, after bringing the four screws to a snug bearing 
again, direct the cross-hair to the edge of some well-defined 
object, as a chain pin, or stake, placed 400 or 600 feet distant. 
Upset the telescope, and place a like mark at about the same 
distance, and level in the opposite direction. Unclamp. Re- 
volve the instrument horizontally on its spindle half way 
round, and direct the cross-hair to the point first observed. 
Again upset the telescope. If the cross-hair now strikes aside 
from the second mark, conect <)ne-(|uarler of the error by 
means of the lateral capstan head seteivs on the barrel, and 



THE TRANSIT. 41 

one-quarter with the tangent screw. Kepeat until the adjust- 
ment is effected. An experienced transitman will generally 
prefer to make this adjustment without aid, points in range 
being readily found. 

5. Having thus brought the cross-hair to revolve in a plane( 
it is next to be seen whether the plane in which it revolves is 
truly vertical. To do so, set the instrument near the base of 
some lofty point, as a church spire or chimney, on which point 
direct the cross-hair, and then, tilting the object end of the 
telescope downwards, set a pin, or make a pencil dot in line. 
Unclamp the spindle; turn the instrument horizontally half- 
way round; clamp fast; fix the cross-hair again on the lower 
point, and try it on the upper one. If it misses, correct half 
the error by means of the adjusting screws now usually pro- 
vided, at one of the bearings of the cross-bar; or, if these be 
lacking, by filing off the feet of the standard which supports 
the higher end of the cross-bar. 

6. TO ADJUST THE NEEDLE. 

Having removed the cap, and placed the instrument con- 
veniently in a still room, push one end of the needle a little 
aside from the point where it tends to settle, and exactly to 
some figured division line on the graduated circle. There 
gently stay it in position by means of a small wooden block, 
an ivory die, or the like. Observe where the opposite end 
strikes. If between graduation lines, mark the precise si^ot 
with a sharp pencil. Turn the needle end for end, and stay 
the reverse j)oint at the division line first observed. Again 
spot with the pencil where the opposite end stakes. Midway 
of these two pencil spots make another. Take the needle off 
the pivot, and bend it this way or that, until, by repeated 
trials, when replaced with one end stayed at the division line 
first observed, the other shall cut the midway pencil spot. 

7. The needle being thus straightened, proceed to rectify the 
position of the centre pin, if necessary, by bending it with nip- 
pers so that the needle shall cut opposite degrees at the quarter 
points of the circle. 



12 



MIS CELLANEO US. 



XV. 



4 



o 
O 

2 " 
1 = 

6 


CO \o <t W O 

II MINI 



MISCELLANEOUS. 



THE VERNIER. 



1. The vernier in the transit is a short 
graduated arc, movable around the graduated 
circle of the instrument, by means of which 
subdivisions of the circle graduation can be 
read. There are many varieties of the ver- 
nier; but a knowledge of the principle upon 
which one is made introduces the student to 
an easy acquaintance with all. 

2. Suppose the tenth part of a foot to be 
marked off on a straight edge into ten equal 
parts, and that on another straight edge a 
space equal in length to nine of these parts is 
divided also into ten equal parts. The sub- 
divisions of the latter scale will then each be 
nine-tenths as large as the subdivisions of the 
former; and if the graduated edges are placed 
together, with the zero marks in both exact- 
ly lined, the first mark of the latter, or ver- 
nier, scale will fall short of the first mark of 
the former, or limb, so to s'peak, by one-tenth 
part of the first space on the limb; that is to 
say, by one-tenth part of one-hundredth of 
a foot, or one-thousandth of a foot. The sec- 
ond mark of the vernier will fall short of the 
second mark of the limb by two-thousandths 
of a foot, and so on. If, therefore, the ver- 
nier scale be moved slowly forward, the suc- 
cessive oppositions of the scale marks will 
indicate successive advances of the vernier, 
each equal to the one-thousandth part of a 
foot. The marginal example reads 6.217 = six 
I'eet, two-tenths, one-hundredth, and seven- 
lliousandths. 

3. The annexed figure represents the transit 



MIS CELLANEO US. 



43 




vernier, together with a part of the graduated circle. The 
vernier is a double one, for con- 
veuieuee in reading angles right 
or left. It will be observed that 
a space, equal to twenty-nine half 
degrees on the limb, is laid off 
from zero each way on the ver- 
nier, and there subdivided, on 
both sides of zero, into thirty 
equal parts. If now the zeros 
are brought into line, tlie first 
marks- of the vernier right and 
left will fall one-thiitieth part of 
a half degree short of the first, or 
half-degree marks on tlie limb; 
that is to say, one minute short. 
The vernier, therefore, is scaled 
to read minutes; and, if its zero 
mark be moved slowly half a 
degree on the limb, its several 
subdivision marks, one after an- 
other in arithmetical succession, 
will be seen to line with marks 
of the limb until the thirtieth is 
reached, when zero will be found 
to have traversed the half degree 
space. 

4. TO READ AN ANGLE. 

First note whether the vernier 
has been moved right or left; 
then observe on the limb the 
number of full degrees, and the 
half-degree, if any, Avhicli zero of 
the vernier has passed; next, 
look along the vernier from its 
zero towai-ds the riglit, if the 
movement lias been towards the 
right, and from zero towards the 
left, if ihe movement has been 
towards the left, until a "minute" mark is found exactly in 
line with some mark on the limb. Add the number of that 



O — 




44 MISCELLANEOUS. 

minute mark on the vernier to the angle already ascertained 
within lialf a degree from the limb: the sum will be the angle 
sought. Tlio vernier in the figure reads 1° 20' L. 

5. In some resjiects a vernier graduated decimally would be 
more convenient on railroad locations, where the 100-feet chain 
is used; the calculation of engineers' tables to sixtieths of a 
degree has prevented its adoption. 

6. TO RE-MAGNETIZE A NEEDLE. 

Lay the north half flat on a smooth, hard surface, and 
v.itli gentle pressure draw the south pole of a common magnet 
over it, from the centre outwai'ds, withdrawing the magnet 
from it six or eight inches after each pass. Repeat ten or a 
dozen times. Treat the south half of the needle in the same 
manner with the north pole of the magnet. Replace the bal- 
ancing wire. If the needle yet proves to be sluggish, take out 
the centre pin, and newly point and polish it. 

7. If the needle, by reason of electricity, clings to the cover- 
ing glass in the field, a touch of the moist finger to the top of 
the cover will release it. 

8. Do not suffer idlers to play it about with knives, keys, 
and the like, 

9. When the instrument is out of use, leave the needle free. 



10. TO REPLACE CROSS-HAIRS. 

Take out the eye-glass tube. Remove the small lateral 
capstan head screws which hold the cross-hair ring athwart 
the barrel. Loosen the vertical screws, and, taking care 
throughout to observe the position of the ring, in order that it 
may be got back again right side up and right face forward, 
turn it lengthwise of the barrel. Insert the end of a pino 
sliver into one of the side holes, take out the vertical screws, 
and withdraw the ring. Stretch across new hairs, in the scores 
traced for them, of the finest clean spider-line; secure thcni 
with a touch of gum or wax, and put the ring in by a rcveise 
process. 

11. TO FIX A TRUE MERIDIAN. 

By equal shadows of the sun. 

On level ground or ice, set up a pole, Two or three hours 



MISCELLAITEOUS. 45 

before noon, mark the extremity of its shadow. With radius 
reaching to that mark, from a centre on the surface vertically 
below the top of pole, strike an arc eastward. Two or three 
hours after noon, watch for the moment when the extremity 
of the shadow touches the arc. There make another mark. 
The true meridian will pass from the centre midway between 
the two^ marks, if the observations be made about the period of 
the solstice, in June or December. The method gives a fair 
approximation at any time of year. 

13. By observation of the North Star in meridian. 

Polaris, or the North Star, being not exactly at the pole, re- 
volves around it through a small 
circle. It is therefore due north • ' ^ 

of an observer only when verti- • • i 

cally above or below the pole. To C A S S i 6 p-p i a 
observe it at either of these points, • 

reference is had to certain bright i 

stars which are in vertical range \ 

with it near the time of culmina- ' 

tion. Its vertical rauge with either Pole ' STAR 

of the reference stars being ob- "^poIe " " 

served, the true meridian may be \ 

set out by means of a direct obser- •. 

vation of Polaris at an interval I 

of time thereafter derived from I 

the accompanying table. • 

The stars thus used are, first, \ * 

the middle star of the three com- D J P P E R • 

posing the handle of the Dipper, ,•*•'' 

called C TJrsm Majoris; second, the t^ 
star called d, at the foot of the • • 
first stroke of the W in the con- \ 
stellation Cassiopeia, which lies I 

opposite the Dipper, at about an \ 



equal distance from the pole. Of course, when one of these 
stars is in upper, the other is in lower culmination; and the 
approximate time for observation may be found in Table I., 
giving the culminations of Polaris. At present, Jan. 1, 1890, 
the Pole Star culminates not quite one minute earlier than it 
comes to the same vertical with C Ursae Majoris— a fact indi- 
cated by the negative sign in the annexed table. The two 
stars will culminate together two and a half jeais lience. 



46 MISCELLANEOUS. 

The following table gives, with sufficient accuracy for any 
latitude in the United States south of Alaska, and for either 
the upper or the lower culminations of these bright stars, the 
value of the time interval, and the annual increase thereof in 
minutes, between the moment of vertical coincidence, and 
the moment of the culmination of Polaris. 

Mill. Min. 

For C Ursm Majoris in 1890 - 0.9 / » , . , n ot. 

in iQnn _i_ 9 ft f Annual mcrease + 0.35 



in 1900 + 2.6 

For 8 Cassiopeia in 1890 + 0.1 ) * , . , a oo 

in 1900 4-3 4 1' Annual mcrease + 0.33 

To establish the meridian, choose still weather, hang a 
plumb-bob from some high fixed object into a bucket of 
water, that it may be both free and steadfast, and select a 
place of observation so far southward that the plumb-line 
shall cover the breadth of sky between the reference star and 
the pole, — the farther the better. The point of observation 
may be an upright bodkin or compass-sight, fastened to a 
block movable horizontally eastward and westward. Watch 
for the moment when, from the point of observation, the 
plumb-line covers Polaris and the reference star. On the 
lapse of the tabular interval thereafter bring the plumb-line in 
range with Polaris by shifting the observation point laterally. 
That range will be the true meridian. Stakes may be set on 
it forthwith by means of candles. 

If the star in Cassiopeia be used within the coming two and 
a half years, attention is directed to the negative time interval. 
Its treatment hardly needs exposition. 

With a transit the plumb-line is not necessary, but special 
care should be taken to adjust the vertical thread of the tele- 
scope, and the horizontality of its transverse axis. This is 
best done by sighting up and down a fine cord or wire sus- 
pending a plummet in water. When making observations at 
night the cross hairs maybe illuminated by reflecting light on 
the object glass from white paper. 

18. By observation of the North Star at its extreme elonga- 
tion. 

Find the time in Table II., and make the preparations above 
directed. Keep the plumb-line in range with the star until 
the star apparently ceases to move. Mark that range. Multi- 
ply the natural tangent of the azimuth, given in Table III., by 



MISCELLANEOUS. 47 

the distance in feet from the point of observation to the mark 
in the northern range just set. The product will be the dis- 
tance from said northern range mark, square right or left, to 
a point in the true meridian passing through the point of ob- 
servation. If the western elongation was observed, set off the 
calculated distance eastward from the northern range mark; 
if the eastern elongation w^as observed, set the distance off 
westward. If both the eastern and western elongations be 
observed, the true meridian will pass through the point of 
observation, bisecting the angle between the northern range 
marks. 

With a vernier instrument, the azimuth can be laid off 
directly, in degrees and minutes. 



i 



PROPOSITIONS AND PROBLEMS 
RELATING TO THE CIRCLE. 



XVI. 

PROPOSITIONS RELATING TO THE CIRCLE 

The following propositions, demonsfrablo by simple processes- 
of geometrical reasoning, may be regarded as axiomatic. 




1. Ill any circle a tangent is perpendicular to radius at tbe 
ta»)g''ut noint. Tims, i> 1 is perpendicular to BC. 

4'.) 



50 PROPOSITIONS RELATING TO THE CIRCLE. 

2. Tangents drawn to a circle from the same point are equal 
Thus, I B = I E. > 

3. The angle DIE, at the intersection of tangents, is equal 
to the central angle B C E, inchuU'd bcLween radii to the tan- 
gent points. 

4. If a chord BE connect the tangent points, the angles 
I BE, lEB, are equal: each of them is equal to half of the 
central angle BCE, or of the intersection angle DIE. 

5. Any angle, BCE, at the centre, subtended by the chord 
BE, is double the angle BFE, at the circumference, on the 
same side of the chord B E. 

6. Acute angles at the circumference, subtended by equal 
chords, are equal. 

7. An acute angle, KFH, between a tangent and a chord, 
is called a tangential angle, and is equal to the peripheral 
angle LFH subtended by an equal chord; each is equal to 
half the central angles FCH, or HCL, subdivided by the 
same chords. 

8. The exterior angle LHN at the circumference, between 
two equal chords, is called a deflection angle : it is equal to the 
central angle, or to twice the tangential angle, subtended by 
either chord. 

9. If F K be made equal to F H, and H X be made equal to 
HL, HK is called the tangential distance, and LN the deflec- 
tion distance. 

10. The exterior angle E HIS" at the circumference, between 
two unequal chords, is equal to the sum of their tangential 
angles, or to half the sum of their central angles. 



XVII. 

CIRCULAR CURVES ON RAILROADS. 

1. The circle is divided, for convenience, into 360 equal 
parts, called degrees. A circle 36,000 feet in circumference 
would be cut by such subdivision into 360 parts, each 100 feet 
long, and subtending an angle of one degree at the centre; its 
radius would be 5,729.6 feet, usually reckoned 5,730 feet. The 



CIRCULAR CURVES ON RAILROADS. 51 

chain 100 feet long being llie unit generally adoiDted by Ameri- 
can engineers for field measurements, any circular arc liaviug 
that radius, of 5,T;jO feet, is called a one-degree curve, for the 
reason that one chain is equivalent to an arc of one degree at 
the circumference. 

2. The circumferences of circles vary directly as their radii; 
hence, in any circular arc struck with half that radius, or 
2,865 feet, one hundred feet at the circumference would sub- 
tend an angle of two degrees at the centre. Such an arc is 
called a two-degree curve. If one-third of the primary radius 
of 5,730 feet, or 1,910 feel, be used, the arc is called a three- 
degree curve; and so on. 

3. It should be borne in mind, however, that these measure- 
ments are supposed to be made around the arc itself, and not 
on lines of chords. Since field measurements with the chain 
are always made on the lines of the choi'ds, which are shorter 
between given points at the circumference than the lines of 
the arcs, as a bowstring is shorter than the bow, it is plain 
that, in advancing tov/ards the centi'e of the one-degree curve 
by a series of concentric circles having radii equal to one-half, 
one-third, &c., of the i)riniary radius, the chord 100 feet long 
differs more and more in length from the arc subtended by it, 
the bow being more and more arched in relation to the string. 
Thus, in the circle having a radius equal to one-twentieth of 
the primary radius, the chord 100 feet long subtends an angle 
of 20° 06', at the centre, instead of 20°, and the arc is 100.5 
feet in length, instead of 100 feet. In order, therefore, that 
the chord of 100 feet may subtend arcs of 1°, 2°, 3°, &c., in 
regular succession, the radii of these successive arcs must be 
somewhat greater than the above method by subdivision of the 
primary radius would make them; though, as might be inferred 
from the extreme case given by way of illustration, the dif- 
ference is not appreciable in ordinary field practice, and radii, 
together with all the functions dependent on them, may 
usually be held to vary as the degree of curvature, or central 
angle per 100 feet chord, varies. 



TO FIND THE RADIUS OF A CURVE. 



XYIII. 

TO FIND THE RADIUS, THE APEX DISTANCE, TH] 
LENGTH, THE DEGREE, ETC., OF A CURVE. 

1. Let D B, A L be two straight lines intersecting at D. Lay 
off equal distances, D A, D B; erect perpendiculars at A and 

B, meeting at G, and con- 
nect A B, D G. From the 
centre G, with radius G A, 
draw the curve A H B. 

The point D will be the 
P. I.; A and B, tangent 
points; D A, D B, the tan- 
gents, or apex distances, 
which denote byAD; D H, 
the external secant, or S; 
HN, the middle ord, or 
O. Let the long chord 
A B, connecting the tan- 
gent points, be called C, 
Call the deflection angle to a 




DBA=AGD=DGB 



and G A or G B, the radius, R 
chord of 100 feet D, as before. 

3. By XVI. 3 and 4, angle DAB 
= iL 

3. GIVEN THE INTERSECTION ANGLE I AND RADIUS R, TO 
FIND THE APEX DIST. AD. 

A D = R X tan. i I. 

Example. 
R= 1,910.1, I = 35° 24'. 

Then A D = R tan. i I =r 1,910.1 X 0.3191 = 609.5. 

4. Measure from the P. I. equal distances, D M, D F, along 
the tangents. Measure, also, MF and D K, the distance from 
D to the middle point of MF. Then, by reason of similarity 
in the triangles M D K, D A G, 

MK:DK::AG:AD::R:T 
.•.AD=RxDK-v-MK. 



TO FIND THE RADIUS OF A CURVE. 53 



Let M K = 190.5, D K = 60.8, R = 1910.1. 

Then R = 1910.1 .... 3.281056 

DK= 60.8 .... 1.788904 

MK= 190.5 (a.c.) '. . . 7.720105 

AD= 609.6 . . . . 2.785065 

5. If 100- feet chords be used, find the ap. dist. in Table 
XVI. corresponding to the given angle I. Divide that tabular 
ap. dist. by the degree of curvature corresponding to the 
given radius: the quotient will be the required ap. dist. Thus, 
Tab. A D corresponding to BS"" 24' = 1,828.7, which, divided 
by 3, the degree of curvature, gives 609.6, the ap. dist. sought. 

6. GIVEN THE INTERSECTION ANGLE I AND AP. DIST. AD, 
TO FIND RADIUS R. 

Transposing the equation in (8), 

R = AD-T- tan. ^ 1= A D Xcot. i I. 

Example. 
4D=609.6, 7=35° 24' R=ADcot. i 7=609.6 X 3.1334=1910.1. 

B}'- a like transposition of the equation in (4), 
R = ADxMK-^DK. 

7. If 100-feet chords be used, find in Table XVI. the ap. 
dist. corresponding to the given angle I. Divide that tabular 
datum by the given ap. dist.; the quotient will be the degree 
of curvature in degrees and decimals. The radius corre- 
sponding to this degree of curvature may be found by (12), by 
Table X., or, with sufficient accuracy for ordinary practice, by 
dividing 5,730, the radius of a 1° curve, by it. 

Thus, in the foregoing example, the tabular ap. dist. cor- 
responding to 35° 24' is 1,828.7. Dividing by 609.6, we have 3 
for the degree of curvature; and 5,730 divided "by 3 gives 
R= 1,910 feet. 



54 TO FIND THE RADIUS OF A CURVE. 



%. GIVEN THE INTERSECTION ANGLE I AND CHORD A B = C, 
CONNECTING THE TANGENT POINTS, TO FIND RADIUS R. 

AG = A N-^sin. AGN ; 
i.e. j B = ^ C -^ sin. i I. 

Example. 
1=35° 24', C= 1161.4. 

Then R = ^C^ sin. 1 1, = 580.7 -^ 0.304 = 1910.2. 

9. If 100-feet cliords be used, find in Table XYI. the chord 
corresponding to the given angle I. Divide that chord by the 
given chord, for the degree of curvature in degrees and deci- 
mals. Determine the corresponding radius by (17), by Table 
X., or, for ordinary practice, by dividing 5,730 by it. 

Thus, in the foregoing example, the tabular chord corre- 
sponding to angle 35° 24' would be 3,484.2, which, divided by 
the given chord, 1,161.4, gives 3 for the degree of curvature, 
and 5,730 divided by 3 makes the radius R = 1,910 feet. 



10. GIVEN THE INTERSECTION ANGLE I AND THE DEGREE 
OF CURVATURE OR DEFLECTION ANGLE D, WITH 100-FEET 
CHORDS, TO DETERMINE THE LENGTH OF THE LONG CHORD 
C, THE MIDDLE ORD. O, THE iLXTERNAL SECANT S, OR 
THE APEX DIST. A D. 

Take from the proper column in Table XYI., the number 
corresponding to the intersection angle, and divide it by the 
degree of curvature: the quotient will be the length required. 

Example. 

A 4P curve, I^ 50° 16'; to find the several functions above 
named. 

Table XVI. gives the designated functions of a 1° curve as 
follows: C 4,867.8, O 543.4, S 599.8, AD 2,688.3. Dividing 
by 4 the degree of curvature, we have for the corresponding 
functions of a 4° curve as follows: C 1,316.8, O 185.6, S 149.8, 
A D 673.0. 



BADU, DEFLECTION ANGLES, ETC. 55 



11. GIVEN C, O, S, OR A D, OF ANY CURVE, AND D, THE DE- 
GREE OF CURVATURE, TO FIND THE INTERSECTION ANGLE, I. 

Multiply the given fuDction C, O, S, or AD, by the degree 
of curvature, D: the product will be found in the proper col- 
umn of Table XVI., corresponding to the required angle. 

Example 1. 
Given A D = 515, D = 5°; to find I. 

Then A D X D = 3,575, which corresponds in Table XVI. 
to 48° 34' = I. 

Example 3. 

Given C = 1,656, D = 3°; to find I. 

Then C X D = 4,968. which corresponds in Table XVI. to 
51° 33= I. 

13. GIVEN C, O, S, OR A D, OF ANY CURVE, AND THE INTER- 
SECTION ANGLE I, TO FIND THE DEGREE OF CURVATURE D. 

Take from the proper column of Table XVI. the number 
corresponding to the given angle I, and divide that tabular 
number by the length of the given part; the quotient will be 
D, in degrees and decimals. 



1. 

Given A D = 587, I = 33° 36'; to find D. 
The A D corresponding to I in Table XVI. is 1,136.3. 

Then 1,136.3 -^ 587 = 1.935 = 1° 56' = D. 

Example 2. 
Given S = 64, I = 30° 25', to find D. 
The Ex. Sec. corresponding to I in Table XVI. is 208. 

Then 208 -^ 64 = 3.25 = 3° 15' = D. 

13. GIVEN THE INTERSECTION ANGLE I, AND DEFLECTION 
ANGLE D, TO FIND THE LENGTH OF THE CURVE. 

Divide I by D: the quotient will be the number of chord 
lengths in the curve. 

If the degree of curvature is a whole number, the more con- 
venient method of effecting the division is, first, to reduce the 



56 



RADII, DEFLECTION ANGLES, ETC. 



raiiiiites, if any, in I to decimals of a degree; then divide bjf 
the degree of curvature. 

Example 1. 
I = 20° 40', 1) = 3°. 20° 40' is equivalent to 20.67 degrees. 
Dividing by :>, we liave 6.89, chord lengths for the length of the 
curve. If the chords, as is usual, are each 100 feet long, the 
length of the curve in this case will be 689 feet. If the chord 
lengths were 50 feet eacli, the length of the curve would be 
half this nunil)er of feet. 

14. If the degree of curvature is fractional, the more con- 
venient method of effecting the division is, first, to reduce 
both I and D to minutes; then divide the former by the latter. 

Example 2, 
I = 30° 22', D = 2° 45'. These are equivalent, respectively, 
to 1,822 and 165 minutes. Dividing the former by the latter, 
we have 1,104 feet for the length of the curve. 

15. The ingenious assistant who will attentively consider 
the preceding figures cannot fail to detect other obvious analo- 
gies which it has not been thought necessary to include in this 
compendium. 

16. In railroad field practice it is usually sufficient to deter- 
mine angles to the nearest minute, and distances to the nearest 
foot. The nicety of seconch and tenths appears generally to 
be quite superfluous; the time consumed on them were better 

employed in pushing ahead. 



17. GIVEN ANY DEFLECTION AN- 
GLE D, AND CHORD C, TO FIND 
RADIUS R. 




FB -^ sin. i AL B = BL ; i.e., 
i C-^sin. i D = B. 

Example. 
Let C=100 feet, D = 4°. 

Then R = ^ C -^ sin. ^ Z> = 50 ^ 
.0349 = 1432.7. 



If the chords are 100 feet long, as is usual in railroad prac- 
tice, radius may be found with sufficient accuracy by dividing 



RADII, DEFLECTION ANGLES, ETC. 57 

5,700, tlio radius of a 1° curve, by the defleetiou angle, or de- 
fjree of curvature. Thus, in the foregoing example, 5,730 -^ 4 
= 1,432.5. 



18. GIVEN ANY EADIUS R, AND CHOPvD C, TO FIND THE DE- 
FLECTION ANGLE D. 

From the preceding equation and example: — 

SUi ^ D = i C -f- K = 50 -^ 1,432.7 = .03 '9 = sin 2° = ^ D 
.-. D = 4°. 



19. GIVEN Any 3i.vi;ius R, and chord C, to find the de- 
flection DISTANCE d. 

First find the deflection angle by above method (18). Then, 
angle HAIi in the figure being made equal to D, and HA 
= B A, BII will be the deflection distance. Draw AK to the 
middle point of H B, 

Then cZ = HB = 2KB = 2AB X sin K A B = 2 C X sm 

Excmiple. 
Let R = 1,14G feet, C = 100 feet. 
By (18) D will be found = 5°. 

Then cZ = 2 C X sin i D = 200 X .0436 = 8.72 feet. 

20. If the chords are 100 feet long, as is usual in field meas- 
urement, divide the constant number 10,000 , by the radius in 
feet: the quotient will be the deflection distance. The deflec- 
tion distance with radius of 10,000 feet and chord of 100 feet 
is one foot: this rule is based upon the principle that deflection 
distances, the chord length being fixed, will vary inversely as 
the radii. 

Thus, in the foregoing example, 10,000 -f- 1,140 = 8.72. 



21. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE TAN- 
GENTIAL ANGLE T. 

The angle T is equal to ^ D by construction; for mode of 
determining it, see preceding section (18). 



58 



ORDIKATES. 



22. G2VEN ANY RADIUS R, AND CHORD C, TO FIND THE TAN 
GENTIAL DISTANCE t. 

First find the tangential angle, as above directed. Then, 
angle B AE in the figure being made equal to T, and AE == 
AB, BE will be the tangential distance. Draw AN to the 
middle point of BE. 

Then t =EB = 2NB = 2AB X sin N A B = 2 C X iin 

Example. 
Let R = 1,14G feet, C = 100 feet. 
By sect. 1, T will be found = 2° 30'. 

Then t = 2 C X sin i T = 200 X .0218 = 4..36 feet. 

23. In ordinary railroad practice the tangential distance jnay 
be considered equal to half the deflection distance. 



I 



XIX. 

ORDIXATES. 

1. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE MID- 
DLE ORDINATE M. 





^r^ 




^ 


E 


*^ 


^>^ 




N / 


K N^ 


/\ 


\ 


\ 


/ 


L 



In the annexed figure, H N = M, H G = R, A B = C. 

NG = VAG2 — AN=2 = VR''^ — iC"; HX = HG — NG, 
i.e., M = R — VR'^ — jC^. 



ORDINATE S. 



Example. 
R = 819, C=100; to find the middle ordinate, M. 

M = 819 — \/07076r=^2500 = 1.53. 

2. Ansle 11 AN = i IIGB; IIG E = ^ AG B, .'. HAN = 
|AGB. 

IIX = AN X ian. HAN; i.e., M = i C X tan. i D; D 
being the central angle subtended by the chord. 

Example. 
D = 7°, C = 100; to find M, the middle ordinate. 

M = i C X tan. { D = 50 X 0.03055 = 1.528. 

3. GIVEN THE EADIUS R, ClIOKD C, AND MIDDLE ORDINATE 
M, TO FIND ANY OTIIEK OllDINATE E K = M', DISTANT d 
FKOM N, TUE MIDDLE I'OINT OF THE CHORD. 

KL = NG;NK=GL;EK = EL — NG. 

E L = VG E-^ — N K^ = VR- — '/' ; NG (1)= Vr- — iC^. 
Then E K = M' = y/W^^' — VR- — i C^. 

4. It is a property of ihe parabola, that ordinates vary as the 
products of their abscissas. Tliis property may be assigned to 
the circle in cases where the arc encloses a small angle. 
Applying it here we have — 

HN : EK :: AN X NB : AK X KB. 

Call any segments A K, K B, of the chord, a and h. 

Then M : M' : : i C'^ : a6, .'. M' = M X 4 a6 -^ C^, 

Example. 
M = 1.528, C = 100, a = 60, ?; = 40; to find M'. 

M' = 1.52S X 9600 ~ 10000 = 1.528 X 0.96 = 1.467. 

5. Multiply the corresponding ordinate of a 1° curve from 
the annexed table by the degree of curvature: the product wili 
l)e the ordinate sousfht. 



ORDINA TES. 



OKDINATES OF A 1° CURVE, CHORD 100 FEET. 



Distances op the Oruinates from the End of'the IOO-feet Chord. 


Middle 
Feet. 

50 


Feet. 
45 


Feet. 
40 


Feet. 

O.J 


Feet. 
30 


Feet. 


FtTt. 
20 


Feet. 
15 


Feet. 
10 


Feet. 
5 


Lengths of the Orihnates in Feet. 


.218 


.216 


.209 


.198 


.183 


.164 .140 


.111 .078 


.041 



Example. 

What is the ordinate of a G° curve, 30 feet from the end of 
the IOO-feet chord? 

The corresponding tahiilar ordinate of a 1° curve is .183; 
which, multiplied by 6, gives 1.09S, the required ordhuitt^ 

6. A quick way of laying otf ordinates on the ground, and 
one sufficiently exact for the Hold, is, after fixing the point II 
by means of the middle ordinate HX, to stretch' a Hue from 
II to A, and make the middle ordinate F O = { II X; then from 
F to A and F to H, making the middle ordinates = ^ F O; and 
so on. 

7. A good track-layer will seldom require points at shorter 
intervals than 25 feet. 



TEACING CUEYES 

AND 

TUKNIISTG OBSTACLES IN THE FIELD. 
XX.— XXIIL 



TRACING CURVES AND TURNING 
OBSTACLES IN THE HELD. 



XX. 



TO TRACE A CUKYE ON THE GROUND WITH THE 
CHAIN ONLY. 



1, This is best taught by an example. 




Example. 
From a point B, 18 feet in advance of A, on tangent A B, to 
trace a: curve of 867 feet radius to the right, with chords 66 feet 
long, and consuming an angle of 34° 27'. 

63 



64 TO TRACE A CURVE ON THE GROUND. 

2. First, dividing half the unit chord, or 33 feet, by thf>' 
radius, 367 feet (XYIII., 18), we have 0.09-|- for the sine of tli. 
:angential angle, corresponding to an angle of 5° 10': the de-: 
flection angle, therefore, Is 10° 20'. The tangential distance! 
corresponding to the angle 5° 10', and chord 66 feet, is equal 
(XVIII., 22) to twice the chord multiplied hy the sine of half 
the tangential angle, = 132 X 0.04507 = 5.95 feet. The deflec- 
tion distance (XYIII., 19) is equal to twice the chord multi 
plied by the sine of half the deflection angle, = 132 X 0.09+ 
^ 11.88, say 11.9 feet. 

3. To find the length of the curve (XVIII., 13): Divide the 
total central angle by the degree of curvature. The central 
angle, 34° 27', is equivalent to 2067 minutes; dividing by 
620, the number of minutes in the deflection angle, we have 
3.33, the number of chord lengths in the curve, = 3^ chains = 
220 feet. 

If A be a stake numbered 2, then the point of curvature, B, 
will be 2.18, and the point of tangent, F, will fall at 2.18 + 
3.22 = stake 5.40. 

4. To determine the tangential distance C P, to the first 
stake on the curve, either of two methods may be used: — v 

First, The sine of any tangential angle is equal to half the 
chord which limits the angle on one side divided by radius. 
The limiting chord B C in this instance is equal to 66 — 18 — 
48 feet; half of 48, therefore, or 24 feet, divided by radius, 367 
feet, gives 0.0654, the sine of 3° 45' =•- tangential angle P B C. 
The sine of half this angle multiplied by twice the given chord 
= 0.0327 X 96 = 3.14 feet, the tangential distance C P. 
6. Secondly, CP may be found as fqllows, assuming that 

the functions of small 
\E angles vary directly as 

the angles themselves, 
and vice versa. 

Let B F be a portion 
of the curve. Make the 
tangent B E equal to the 
chord B F, and strike the 
arc E F. Draw the sub- 
chord B C, and strike the 
arc C P. Prolong B C to D. E F may be taken as the tangent 
tial distance due to the whole chord BF, and PC the tangen- 
tial distance due to the sub-chord B C. 




TO TRACE A CURVE ON THE GROUND. 65 

Then PC : ED : : B C : BD or BF; and, by the foregomg 
supposition, E D : E F : : B C : B F. Combining these propor- 
tions, and oancelling E D, we have P C : EF : : B C'^ : BF- .'. 
PC = EF X (BC-^BF)2. 

In words, the tangential distance for a sub-chord is to that 
for a whole chord as the square of the sub-chord is to the 
square of the whole chord. The same is true of detlection dis- 
tances. 

6. In the example we are treating, the tangential distance for 
the whole chord of CO feet has been found to be 5.95 feet; 
that f jr 48 feet, therefore, is 5.95 X 48- -f- 66- = 5.95 X 0.528 
= :'•>. 14, as before. 

Stretch the 48 feet of chain from B to P, in prolongation of 
tangent A B, and ina:lc the point P ; thc:i stop aside, and stretch 
from B to C, making the distance PC = 3.14 feet: C will be a 
stake on the curve. 

7. Next, run out the whole chain length from C to O in the 
range BC. To find CD, suppose the line jS" C T to be drawn 
tangent to the curve at C. Then ND may be considered the 
tangential distance due to the whole chord, ^= 5.95, as above 
determined. 

The angle OCN = TCB=PBC (XVI., 4); and (5) 

ON:ND::BC:CD.-.ON = NDxBC-v-CD, i.e.;OD 
= N D 4- O N = N D -f- N" D X (B C -^ C D) = 5.95 X 1 + (48 
-i- 66) = 5.95X1.727 = 10.27. 

8. The point N may be fixed otherwise by laying off B T = 
C P, and running out the chain length C N in the range C T. 
The point D on the curve may then be fixed by making N D 
equal to 5.95 feet, the tangential distance. 

Next run out the chain to M, in the range C D ; make M E 
equal to the deflection distance, 11.9 feet, and fix the point E. 
The points C, D, and E will be stakes 3, 4, and 5 on the curve. 

9. To set the point of tangent, F, at stake 5.40, prolong the 
chord line D E for 40 feet to L, and suppose Y E to be drawn 
tangcRt to the curve at E. Then the angle LEV is equal to 
the tangential angle of the curve; and the sub-tangential dis- 
tance L V is to the wdiole tangential distance due to the 66- 
feet chord, as the sub-chord is to the whole chord (5); i.e., 
L V = 5.95 X 40 ^ 66 = 3.6 feet. 

By the method JUustrated in (6), the distance FV will be 



66 TO TRACE A CURVE ON THE GROUND. 

equal to 5.95 X 40^ -^ 662 = 5.95 x 0.367 = 2.18 feet. W}*'i». 
the distance LF = 3.6 + 2.18 = 5.78 feet, thus obtained, and 
the sub-chord E F = 40 feet, the point of tangent F may be 
established. 

10. Next, set off UE = FY = 2.18 feet, and lay out FK in 
prolongation of the range U F ; F K will be in the line of the 
terminal tangent. 

11. This analysis has been somewhat minute and detailed, 
in order that the subject may be thoroughly understood. An 
instrument for measuring angles should always be used in rail- 
road service: it greatly simplifies and abridges the labor of 
tracing field-curves, and gives more exact results. But occa- 
sions sometimes rise, in miscellaneous practice, when strict 
accuracy is not required, and the chain only can be had: the 
young engineer should qualify against such occasions. 






XXI. 

TO TRACE A CURVE ON THE GROUND WITH 
TRANSIT AND 100-FEET CHAIN. 

1. This, also, is best taught by an example. 

Let it be a general rule, in locating, to fix the intersection of 
tangents, and to set the tangent points, or the P. C. at least, 
from the P. I. There are exceptional conditions, as a steep 
hillside, timber or broken ground, a very long arc, unimpor- 
tance of exact conformity to the project, and the like, which 
warrant its omission; but where these conditions do not obtain 
or are not prohibitory, and a snug fit is desirable, time will 
usually be saved by fixing the P. I. It often proves serviceable 
as a reference point during construction: on the location, i1 
gives confidence in the work and an assurance of safe progress, 
which are well worth a little painstaking beforehand. 

2. Having established the P. I., and found the intersection 
angle to measure, say, 66° 45^ the first step is to find the apex 
distances so called, or tangent lengths IB, IF. These are 
each equal to R X tan. 1 1. If .a 7° 30' curve be prescribed to 
close the angle, R X tan. ^ I = 764 X 0.659 = 503 feet. 



TO TRACE A CURVE ON THE GROUND. 



61 



Or, referring to Ta- 
ble XVL, theap. dist. 
corresponding to 66° 
45' is found by inter- 
polation to be 3774.6; 
dividing by 7.5, the 
rate of curvature in 
degrees and decimals, 
we have for the apex 
distance 503 feet, as 
above. 

3. Before disturbing 
the instrument, which 
is presumed to stand 
in the range of the 
terminal tangent, 
measure I F, = 503 
feet, and set the P. T. 
at F. Then direct the 
telescope to the last 
point fixed on the ini- 
tial range AB, meas- 
ure I B, = 503 feet, 
and set the P. C. at B. 
Move to B. 

4. Suppose the P. C. 
to have fallen at a 
stake 2.50. In order 
to find the length of 
the curve, divide the 
intersection angle by 
the degree of curva- 
ture, having first re- 
duced the minutes in 
each to hundreths of a 
degree by multiplying 
by 10 and dividing the 
product by 6. Thus 
the intersection angle 
becomes 66.75°, and 
the degree of curva- 
ture 7.5° : dividing the 




6S TO TRACE A CURVE ON THE GROUND. 

former by tlie latter, we have 890 feet for tlie length of the 
curve. 

Or, the intersection angle 60° 4o' is equivalent to 4005', and 
the degree of curvature 7° 30' is equivalent to 450': dividing 
the former by the latter, we have 890 feet for the length of the 
curve, as before. 

5. Adding 8.90 to 2.50, the number of the P. C, the P. T. Is 
found to fall at stake 11.40. Let the rear chainman make a 
note of this, that there may be no mistake in the terminal pluf^. 

6. Next, to determine the proper deflections from the line of 
tangent at B, bear in mind that the deflection for a whole 
chain is half the degree of curvature ; and that, in field-curves 
of more than 300 feet radius, the deflections for sub-chords, 
or plusses, may, without material error, beheld to vary directly 
as the sub-chords themselves; that is to say, the sub-deflec- 
tions due to 30, 60, and 80 feet, for instance, will be, to the 
deflection due to 100 feet, as 30, 60, and 80 are to 100. 

7. Thus, in the example, 7° 30' being the degree of curva- 
ture, one-half of this, or 3° 45', will be the deflection due to a 
chord of 100 feet; and ^sjl of this, or a deflection of 1° 52^' 
from the line of tangent at B, will fix stake 3, 50 feet distant 
on the curve. 

8. The following is a simple rule for finding sub-deflec- 
tions: — 

Multiply the sub-chord in feet by the rate of curvature in 
degrees and decimals : three-tenths of the product will be the 
sub-deflection in minutes. 

Thus, in the example, 50 X 7.5 = 375, and 375 X 0.3 = 
112.5' = 1° 52i', as before. 

9. Having set stake 3, stakes 4 and 5 will be fixed by succes- 
sive deflections of 3° 45'. In establishing stake 5, the index 
will read, 1° 52^' -|- 3° 45' -f 3° 45' = 9° 22f = angle C B 5. 

10. Suppose. the instrument moved to 5. See that the ver- 
nier has not been disturbed, backsight to B, and deflect 9° 22^' 
right; i.e., double the index angle. The index will now read, 
18° 45' = the angle I CD; and the telescope will be directed 
along the line C D, tangent to the curve at 5, for the reason 
that the angle B5C has been made equal to the angle CBS 
(XYI. 4). 

Proceed with successive deflections of 3° 45' from this tan- 
gent, and set stakes 6, 7, 8, and 9, at intervals of 100 feet. 
U. Suppose the iustrumeut moved to 9. lu fixing this 



TO TRACE A CURVE ON THE GROUND. 69 

stake, the index will read, 18° 45' + 4 times the constant angle 
;io 45/, = ISO 45' -f 150 = angle I C D + angle D 5 9, = 33° 
45'. In order to place the telescope in the line D E, tangent to 
che curve at 9, it is now necessary to turn an angle to the 
right, from backsight to 5, equal to D95 = D59 = 15°; i.e., 
the vernier should be moved from 33° 45' to 33° 45' -f 15° = 
48° 45'. The telescope will then be in tangent at 9. 

12. A simple rule for tinding the hidex angle which shall 
place the ti;lescope in tangent at any point on the cur\ e is as 
follows: — 

From double the index angle which fixed the given point, stih- 
tract the index reading in tangent at the last turning-point : the 
remainder will he the required index angle. 

Thus the index angle which established stake 9 was 33° 
45'. Double this angle will be 67° 30'; subtracting 18° 45', the 
reading in tangent at the last turning-point, we have 48° 45', 
the required index angle, as before. 

The reasons for the rule will be obvious from an examina- 
tion of the figure. 

13. Being in tangent, then, at 9, and the index reading 48° 
45', a deflection of 3° 45' will fix 10: a further deflection of 3° 
45' will fix 11, and the index will stand at 48° 45' -f 7° 30' = 
56° 15'. 

14. To find the deflection corresponding to the sub-chord 11 
F, =40 feet: by the foregoing rule (8), the degree of curva- 
ture, 7.5, multiplied by 40, the length of the sub-chord in feet, 
gives a product of 300, three-tenths of which amount to 90 
minutes = 1° 30'. Adding 1° 30' to 56° 15', makes the index 
angle 57° 45' to fix the P. T. at 11.40. 

15. Move to the P. T. at 11.40, see that the vernier has not 
been disturbed, and backsight to 9. By the foregoing rule 
(12), double the index angle, 57° 45', less the angle in tangent 
at 9, the last turning-point, 48° 45', = 115° 30' — 48° 45', = 
66° 45', = the index angle in tangent at the P. T., = the tota] 
angle consumed by the curve. The work thus proves itself. 

16. The preceding example would appear in the field-book 
as follows: — 



10 



TO TRACE A CURVE ON THE GROUND. 





















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TO TRACE A CURVE ON THE GROUND. 



71 



17. This mode of running curves secures a record of each 
step in the proceeding; so that, if any error occurs, it can 
readily be detected. At each turning-point, the number in 
the " tangent'' column must correspond with the central angle 
due to the length of curve to that point ; and at the P. T. that 
number must correspond with the total central angle. The 
work can thus be checked with facility during its progress, 
and checks itself at the end. 

18. The young transitman is recommended to rule blanks 
after the pattern given, and exercise himself thoroughly in 
computing the parts, and recording the field-notes of various 
curves assumed at will : drawings are not necessary. 

19. Another method, and in some respects a better one, is, 
before starting on a curve, or during its progress, to record for 
all its stations the deflections which would locate them if the 
instrument remained at the P. C. Obviously the final deflec- 
tion thus recorded would be half the central angle of the 
whole curve; and, if the instrument were placed anywhere on 
the curve, a B. S. to P. C, and deflection of half that central 
angle would locate the P. T. The same reasoning will apply 
to any subdivisions of the curve which may be found con- 
venient in field work, the deflection angles for plusses being 
reckoned from the P. C. and recorded as in the case of whole 
stations. I am indebted to Mr. Robert Burgess, C.E., for 
recommending this method. Following is his illustration of 
it as applied to the preceding example. This illustration 
serves also to exemplify another form for field-notes. 



Sta. 




Dep. 

Angle. 


Total 
Angle. 


Calc. 
Curve. 


Mag. 
Curve. 


Remarks. 


12 


O / 


o / 


o / 


o 


o / 




-^40 


X P.T. 


33 22^ 


66 45 


N. 15 E. 


N.1510E. 




11 




31 52i^ 








Apex Dis. 503 


10 




28 071^ 








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20. The transitman, at P. C. , Sta. 3 -f- 50, sets his vernier at 
zero, takes his B. S. and locates the stations in order to 5, 
where a point is given. He then moves to 5, sets vernier at 



72 TO TRACE A CURVE ON THE GROUND. 

zero and backsights to P. C. A deflection of 9° 22^' will now 
place him in tangent; a deflection of 13° 07^' will locate Sta. 
6, and so on, using the recorded angles. Suppose a tree to 
mask the site of Sta. 8. Placing a stake as near thereto as 
possible, to hold distance; go on deflecting to 9, the vernier 
reading 24° 22^', where another point is given. Moving now 
to 9, the transitman observes the general rule, applicable to this 
method, namely, to set his vernier at each new instrument point, 
to the angle recorded opposite the point of his proposed backsight. 
In this case that point is Sta. 5. He therefore sets the vernier 
at 9° 22i', backsights to 5, and turns to 20° 37i' to locate 8. 
The vernier at 24° 22^' would put him in tangent, and the suc- 
cessive recorded deflections complete the curve. The transit 
is then moved to P. T., the vernier set at 24° 22^', the recorded 
angle opposite last point at 9, according to the above rule, 
backsight taken on that point, and a final deflection to 33° 22^' 
turns off the tangent ahead. 

21. The chief advantages of this method are the complete- 
ness of the record for subsequent use; its adaptation to a re- 
tracing of the line backward as well as forward, which please 
observe particularly; the little labor it imposes in mental arith- 
metic; and its simplicity, permitting any one who can read 
figures and turn an angle to relieve the transit on occasion. 

The writer was raised on the first method; is still partial to 
it, out of a certain loyal feeling to the elder generation from 
whom it descended to him; but it must be owned the children, 
in some things, have surpassed the fathers,— as needs they 
should, else were there no progress, — and this seems to be one 
of those things. 



TURNING OBSTACLES TO VISION IN TANGENT. 



73 



XXII. 



TURNING OBSTACLES TO VISION IN TANGENT. 

1. Draw CF parallel to AB. Let lines BC, CE, FG, cut 
these parallels at equal 
inclinations. Call this 
angle I. Then B C = 
CE = FG. BE = 
BD + DE = 2BD. 
But BD = BC -COS. I, .-. BE = 2 BC cos. L EG = CF. 
BG = EG + BE = CF + 2BC cos. I. 




Example. 
Suppose B to be a stake 24.50 on the tangent AB, and that 
a deflection left of 10° be made there for 200 feet to a point C. 
Set transit at C, vernier reading 10° left. B S to B, and deflect 
20° right. Vernier will now read 10° right, and telescope w ill 
be in line C E. Make C E = 200 feet. Move to E. See that 
vernier still reads 10° right. B S to C, and turn 10° left. Ver- 
nier will now read zero, and telescope will be in line E G, or 
tangent AB prolonged. 

Distance BE = 2 B C cos. 1 = 2 (200 cos. 10°) =400 X .985 
= 394 feet. Then E = 24.50 + 394, = stake 28.44 on tangent 
A B prolonged. 

If a parallel line C F were run, a deflectioij of 10° right would 
be made at each of the points C and F. If C F were 250 feet, 
then B G would be = 250 + 394 = 644 feet, and the point G 
would fall at stake 30.94 on tangent A B prolonged. 

2. If angle I = 60°, the other conditions of above method 
being observed, triangle BHE will be equilateral, and BE = 

B H = H E. If the parallel D C 
or DF be run, BE = BD -f 
DC, and BG = BD + DF. 
For field work see last example. 
3. In turning obstacles by 
either of these methods, the 
angles should be measured with extreme niceness. Handle 
the instrument lightly, to avoid jarring the vernier; and, if 
possible, observe well-defined distant objects in the several 
short ranges, that the Une§ of foresight ^n4 backsight may 
accurately coincide, 




74 TURNING OBSTACLES TO MEASUREMENT IN TANGENT. 

In locating, the following method is preferable to those given 
above, and should always be used on long tangents. 

4. Having established points A and B on the centre line, 
the farther apart the better within limits of distinct vision, 
set off the equal 



rectangular d i s - "] 

tances AE, B F, ■ 

ranging clear of a 

the obstacle. 

Place the transit at E or F, fix points G and H on the forward 

range, and, rectangularly to these points, establish others on 

the forward range of the centre line at C and D. The offset 

distances should be measured very carefully with the rod, or 

with a steel tape if they exceed in length the pocket rule 

which every engineer should have about him. 



XXIII. 

TURNING OBSTACLES TO MEASUREMENT IN 
TANGENT. 

1. Fix a point on tangent A B prolonged at E. Lay off at B 
a perpendicular of any convenient length. Move the instru- 
ment to D, make the angle B D A 
= B D E, and mark the point of 
intersection A. By reason of 
symmetry in the triangles A D B, 
B D E, A B = B E, and may be 
measured on the ground. 

2. Or, fix the point E, and lay 
off the perpendicular BD as before. Move to D, direct the 
telescope to E, turn a right angle EDO, mark the point of 
intersection C, and measure C B. Then, by reason of simi- 
larity in the triangles CBD, DBE, CB:BD::BD:BE, 
.-. BE = BD2-i-BC. 




Example. 

Suppose B D to be 60 feet, and B C 40 feet. Then B D 2 -^ 
B C = 3600 ^ 40 = 90 feet = B E. 

3. Or, with the instrument at D, measure the angle BDE. 
Then B E = B D tan. BDE. 




TURNING OBSTACLES TO MEASUREMENT IN TANGENT. 75 

Example. 
B D = 120 feet, angle B D E = 54° 40'. B D tan. B D E == 
120 X 1.41 = 169.2 feet = BE. 
4. Or, without an instrument, lay off any convenient lines B F 
or C H. Mark the middle point 
D. Line out H Gr, parallel to 
AB. Mark on it the point G 
in range with D and E. Then 
GF = BE, orGH = CE. 

5. Should the use of a right 
angle be inconvenient, turn any 
angle E B D = x, measure B D 
about equal by estimation to B E, if the ground permits, and 
set a point D. Move to D, and measure angle B D E =7 2/. 

Then the angle B E D, or z, ^ 180 — ix-\-y), and, by trigonom- 
etry, sin. z : sin. y : : B D : BE, . •. B E = B D sin. y -^ sin. z. 

Example. 
Let x=A4P 02', y = 71° 48', B D = 300 feet. 

Then z = 180° — {x -\- y) = 180° — (44° 02' + 71° 48') = 
180° — 115° 50' = 64° 10'. BE = B D sin. y -^ sin. z = 300 
sin. 71° 48' -f- sin. 64° 10' = 300 X .95 -^ .90 = 316.6 feet. 

The calculation by logarithms would be as follows: — 

Log. 300 2.477121 

Log. sin. 71° 48' 9.977711 

Sum 12.454832 

Log. sin. 64° 10' 9.954274 

Log. 316.6. Diff 2.500558 ' 

n E is invisible from B, extend the line D B towards C, 
until a line C E clears the obstacle. The point E must then 
be established by intersection of the sides CE, DE, in triangle 
C D E. Supposing the extension B C to have been 120 feet, 
the side CD will be 420 feet, the angle y 71° 48'; and, by a 
calculation similar to the above, the side D E, opposite angle x 
in the lesser triangle, identical with DE in the larger one, will 
be found to be 231.7 feet. The sum of the angles at the base 
C E of the triangle C D E = 180° —y = 180° — 71° 48' = 108° 
12'. By trigonometry, two sides and the included angle being 



76 TURNING OBSTACLES TO MEASUREMENT IN TANGENT. 

known in any plane triangle, the sum of the known sides is tc 
their difference as the tangent of the half sum of angles at base 
is to the tangent of half their difference. In triangle C D E, 
therefore, CD + DE or 651.7 : CD — DE or 188.3 :: tan. 
108.12 -!- 2 or tan. 54° 06' : tan. 54° 06' X 188.3 -^ 651.7 = 
.399 = tan. 21° 45', = half the difference of the angles at the 
base. 

Log. 188.3 2.274850 

Tan. 54^06' 0.140334 

Sum 2.415184 

Log. 651.7 2.814048 

Tan. 21° 45'. Diff 9.601136 

The angle at C, being evidently the lesser of the two angles 
at the base, is equal to the half sum of these angles decreased 
by their half difference, = 54° 06' — 21° 45' = 32° 21'. 

Set the transit, then, at C, foresight to D, deflect 32° 21' left, 
and fix in that range two points F and G, between which a 
cord may be stretched, and as nearly as can be judged on 
opposite sides of E. Move to D, foresight to C, deflect 71° 48' 
right, and establish a point E at intersection with FG. Cross 
to E, B S to D, and deflect the angle z = 64P 10' into the line 
of the tangent A B prolonged. 



' 



SUGGESTIOISrS AS TO FIELD-WORK 
AND LOCATION -PROJECTS. 

XXIV. -XXV. 



SUGGESTIONS AS TO FIELD -WORK 
AND LOCATION -PROJECTS. 



XXIV. 



SUGGESTIONS CONCERNING FIELD-WORK. 

1. The Chief Engineer, after conference with his em- 
ployers in regard to the character of the work contemplated, 
and its general route, should, before organizing field-corps, go 
over the ground in both directions, and, aided by the best 
attainable maps, qualify himself by actual observation to in- 
struct his assistants as to the conduct of the survey. Equipped 
with hand-level, pocket-compass, and in rough regions with 
the aneroid, he can often not only prescribe lines for examina- 
tion, but indicate the gradients to be tried, thus saving a vast 
amount of random labor and needless expense. Such thorough 
preliminary exploration is due both to himself and his princi- 
pals: it is too often omitted, or done with a perfunctory rush. 
In broken topography, no maps, notes, or information derived 
from others can supply the want of personal acquaintance with 
tlie ground itself. He must indispensably make that acquaint- 
ance, in order to project an intelligent location, — a work which 
should rarely be delegated; being capital service, it comes 
within the special function of the chief engineer, and only the 
necessary distribution of labor attending a great charge should 
relieve him from its direct performance. 

2. A Field-Corps in settled regions generally consists of 
one senior assistant or chief of corps, one transitman, one 
leveller, one rodman, two chainmen, one slopeman, and two 
or more axemen. 

Tlie following notes in regard to the allotment of duties and 
tlie conduct of work may be acceptable. They are copied 
from the writer's memoranda for the guidance of his field- 
parties, with the addition of some detail, and practical hints 
here and there, to aid the inexperienced. 

79 



80 SUGGESTIONS CONCERNfJSrG FIELD -WORE. 

3. The Senior Assistant will receive instmctions from 
the principal assistant in charge, or the chief engineer, and 
will act exclusively under his direction. 

He will be held responsible for the good conduct of the corps» 
and for the rapid, exact, and economical performance of the 
work. Indecent or blasphemous outcries in the field should 
be prohibited. The writer's various travel by land and sea 
has brought him acquainted with many cultivated, estimable, 
energetic, profane fellows, but not one in whom swearing was 
a grace ; nor has he ever seen an- instance where it forwarded 
work. Those considerate of others' pride and self-respect 
will generally find that a good leader makes good followers. 

The senior assistant is empowered to appoint and dismiss 
employes below the rank of rodman, and will report any 
inefficiency or neglect of duty in the ranks above to his 
chief. 

He will pay the authoi-ized expenses of the corps for sup- 
plies, repairs, transportation, and subsistence, taking duplicate 
vouchers. Accommodations should be sought near the work. 
When not thus obtainable, transportation to and from the 
field is to be regarded as a measure of economy for the com- 
pany, compensating the expense incurred by saving time and 
labor. 

He will superintend field operations in person, keeping in 
advance of the transit to direct and expedite the work, and 
establish the turning-points. On preliminary surveys, the axe 
should be little used ; and on alternative locations, or such as 
may be subject to revision, trees over four inches in diameter 
need rarely be felled. 

He should be patient with sensitive landholders. He will 
find exercise for that amiable virtue, also, with the field vis- 
itors who so often spare time from useful toil to tell him he is 
on the wrong line, and to show him where the right one is. 

Note for record the kind and quality of material to be moved, 
observing quarries, wells, or other indications for the purpose; 
the timber and rock in the country traversed, with a view to 
their use in construction, and the widths of passage to be pro- 
vided for streams, together with the character of their banks 
and beds. 

Note the names of residents in the immediate vicinity of the 
work on survey; and, on location, cause the property-lines to 
be observed and recorded also when convenient. 



SUGGESTIONS CONCERNING FIELD-WORK. 



81 



Always begin grade-lines at the summit, and work down. 
For such service, carry habitually a slip of profile paper, say 
six inches wide and two feet long. Rule the proposed grade- 
line on it, assume a summit cut, mark the stations, and start 
down. When at fault, the elevation can be spotted on the 
profile, which will show at a glance, without any calculation, 
how you stand in relation to grade. 

The work of each day should be compiled and recorded in 
the evening, that no delay may result from the loss or deface- 
ment of a field-book. 

FORM FOR SURVEY RECORD. 



Sta. 


Dis. 


Deflec. 


Course 


M.C. 


Eleva. 


Slope. 


Remarks. 



















FORM FOR LOCATIOX RECORD. 



i 
t 


i 

IS, 


6 


i 
i 


i. 
o 

< 
> 


5 


u 
P 

t 
O 


i 

< 

< 
> 




Remarks. 























On location, check the transitman's calculation of the length 
of each curve and the fractional deflections. 

The senior assistant must be qualified to locate a line accu- 
rately on the ground from the project furnished him. Lateral 
deviations exceeding five feet on ten-degree slopes, three feet 
on fifteen-degree slopes, and two feet on twenty-degree slopes, 
will be considered errors requiring correction. Measurements 
to the experimental line should be made and noted frequently, 
in order not only to check the field-work, but that the line 
may by means of them be laid down on the map. 

The senior assistant will supply himself with drawing in- 
struments, colors, brushes, and the like personal furniture of 
an engineer. He will take care also that the stationery, field- 
books, instruments, and other articles of outfit suiiplied by the 
company, are not misused. His field equipment should always 
include a hand-level and a pocket-compass: to these may be 



82 SUGGESTIONS CONCERNING FIELD -WORK'. 

added a straight, round staff, five or six feet long, steel pointed ^ 
it will be found exceedingly useful. 

If without a topographer, he should make sketches of irregu- 
lar ground, of streams, buildings, roads, and the like, to help 
in compiling the map. 

In hilly or wooded districts, the front chalnman carries the 
flag on survey, and is at the head of the line. In open, plain 
country^ work is greatly forwarded by detaching an axeman 
with flag, to accompany the senior in advance, and set turning- 
points for the transit. The transitman follows as rapidly as 
possible, and the chainmen come after, lining in their stakes 
by the eye from point to point. The whole force is thus kept 
pretty steadily in motion. 

On wide plains, a set of chain-pins may be used, and survey- 
stakes placed five hundred or a thousand feet asunder. Yery 
often stakes at intervals of two hundred feet are sufficient, the 
levels being taken every hundred. Location stakes are put in 
every hundred feet. 

4. The Transitman will be expected to keep his instru- 
ment in adjustment, and to be quick and accurate in its manip- 
ulation. It is not needful to plant it as if for eternity. On 
the contrary, it should be set gently, the legs thrust but slightly 
into the ground, and the screws worked without straining. 

On long tangents it is a good plan to reverse the instrument 
at each new point, putting the north and soiitli ends forward 
alternately. Small errors in adjustment are thus balanced in 
some measure. Select also, in such a case, some distant object 
in range, when practicable, to run by. The telescope, in wind 
or sun, will sometimes warp a little out of line. 

Never omit to note both the calculated and magnetic bear- 
ings of the- lines on survey, and of the tangents on location. 
Guard against the error of reading deflections or bearings from 
the wrong ten mark; as, for instance, 34 instead of 26. 

At the beginning of a curve, let the rear chainman know the 
plus of the P. T. Tell the front chainman the degree of curve, 
and instruct him how, by multiplying 1.75 by the degree, he 
can find the distance of each full station from the range of the 
last two. A quick fellow will soon pick this up, and become 
wonderfully skilful in practice. Thus accomplished, he is a 
check on wrong deflections. 

In running curves, a tangential angle of fifteen degrees from 
one point should seldom be exceeded : twenty degrees is to be 
regai'ded as a maximum. 



SUGGESTIONS CONCERNING FIELD-WORK. 83 

Carry a pocket-compass, and observe with it the magnetic 
bearings of streams and roads crossed. 

Record daily each day's run; fill out the distance column, 
ti'anscribe the chain-book, and, on location, record the apex 
distances also in the column of remarks. 

On survey, do not erase from the field-book the notes of 
abandoned lines. Simply cancel, and mark them " aban- 
doned," in such manner that they may still be legible. 

When required by the senior assistant, the transitman will 
aid in the making of maps. 

5. The Leveller must be familiar with the adjustments 
of his instrument, keep it in order, and handle it rapidly. 

On survey, establish and mark benches at half-mile intervals, 
on location, four to the mile when practicable. 

Note the surface elevations, the depths, and the flood heights, 
of all considerable streams crossed. Take elevations in the 
beds of small streams. 

Six hundred feet each way should be regarded as the maxi- 
mum sweep of the level. 

Carry a hand-level, and thus save the time required to peg 
across narrow hollows, or over heights which can be turned 
with the instrument. 

The leveller should record his work, and make up the profile 
daily. 

6. The Rodman will give his intermediates close by the sta- 
tions, observing the number of each one as a check on the 
chainmen, and calling it out to the leveller. He should have 
an eye to abrupt irregularities in the ground, and give plus 
elevations when necessary. 

He will keep note of bench-marks and turning-pegs, describ- 
ing the latter occasionally with reference to the nearest stake, 
that the levels may be taken up speedily in case of a revision 
of the line. 

When unaccompanied by an axeman, the rodman is equipped 
with belt and hatchet. Sometimes he is furnished also with a 
steel pin for turning on. The pin has a ring through the head, 
by which it may be hung to a spring hook in the belt. 

The rodman will assist the leveller at record and profiles, 
and transcribe the slope-book daily. 

If stakes of survey are set at intervals of two hundred feet 
give rods every hundred feet, as nearly as the midway point* 
can be guessed. 



84 THE CURVJE -PROTRACTOR. 



slopes for one liundred feet on eacli side of the line at every 
station. 

8. The Rear Ciiainman" will carry a book ia which to note 
the turning-points, the crossings of roads, streams, swamps, 
woodland, and, when convenient, i^roperty lines also. He will 
hand it in daily to the transitman for record. As each succes- 
sive chain is stretched, the rear chainman calls out the number 
of the stake it is stretched from: this assures the selection of 
the right number for the stake ahead. 

9. One Axeman will be detailed to make stakes, another to 
mark and drive them. Additional axemen may be employed 
at the discretion of the senior assistant, as the work requires 
them. Wanton destruction of timber, fences, growing crops, 
or other property, should not be allowed. Axemen must be 
careful, in passing through the country, to do as little damage 
as possible. 



XXV. 

THE CURVE-PROTRACTOR, AND THE PROJECTING 
OF LOCATIONS. 

1. The curve-protractor is simply an eight-inch, semi-circu- 
lar horn protractor, upon which a series of twenty-three curves, 
from half a degree up to eight degrees, is finely engraved, to a 
scale of 400 feet to an inch. After some years' use in his own 
practice, the contrivance was transmitted by the writer to the 
well-known firm of James W. Queen & Co., mathematical- 
instrument makers. New York and Philadelphia, by whom it 
is now manufactured. It greatly facilitates the projecting of 
lines and solution of field-problems on location. It enables 
the engineer, for example, by a short, graphical process and 
rapid inspection, to find the curve which shall close an angle 
between tangents, or terminate a compound curve, and pass at 
the same time through some fixed intermediate point, without 
liability to the errors, and free from the loss of time, involved 
in a tedious calculation. Other applications, such as the nice 
adjustment of line among buildings, on precipitous steeps, and 
the like, will suggest themselves to the experienced reader. 



THE CURVE -PROTRACTOR. 85 

2. For office use, the writer prefers a home-made curve- 
protractor of mica, prepared as follows: Take a thin, clear 
sheet, say six by ten inches, free fron\ bubbles and cracks. 
Block it securely on the drawing-table with thumb-tacks, set- 
ting the shanks close against the edge of the sheet, but not 
piercing it, and the heads lapping its edge. From a centre, 
midway of one of the long sides and near its margin, strike the 
curves from 12° or less, varying outwards by half-degrees to 
0°; thence by quarter-degrees to 4°; and thence by ten-minute 
differences to 2^°. This covers one side of the sheet, the scale 
being 400 feet to an inch. ?^ow release the sheet, turn it over, 
and on its other face strike the remaining curves, down to 
ten minutes, from centres on the table, in the reverse direc- 
tion, so that they shall cross the first series at a large angle. 
Space them about three-eighths of an inch asunder at the mid- 
dle. Use a needle-point centre for the first series, to avoid 
boring a large hole in the sheet! Add also, on that face, two 
radial lines drawn towards the corners. Score the fractional 
<3urves very lightly, the full figure curves a little deeper, but 
all of them with steadiness and delicate stress. Practise 
beforehand on a separate slip, for the right intensity of stroke. 
Engrave the numbers with a stiff steel point on the opposite 
side of the sheet to that upon which the corresponding curve 
is traced. Bring the work out by rubbing it with India ink. 
If preferred, the flat curves on the reverse side may be colored 
with carmine. Duplicate protractors will be found useful in 
projecting compound curves. Clip off the four corners, re- 
enforce the edges with a narrow ribbon of tracing-linen, folded 
over them and glued fast, and the article is complete. It is 
perfect for its use; durable, flexible, spotlessly transparent, 
not liable to warp or change dimensions with changes in the 
temperature or moisture of the air, an,d, withal, takes and pre- 
serves a visible line, thin as the gossamer. 

3. To experienced locating engineers, the curve-protractor 
needs no wordy commendation. Contrasted with the incon- 
venient appliances of the old method, — cardboard, veneer, 
glass, or dividers, — its advantages will be ma-nifest. A few 
hints as to the manner of using it may be in place. 

4. First of all, let the experimental line approximate to the 
probable line of location; and, upon that base, construct a 
contour map, with reference to which special observations 
shou.ld be made in the field, and the chaining done with care. 



86 THE CURVE -PROTRACTOR. 

Extreme accuracy in the contours need not be attempted. 
Note the courses of streams, ravines, and ridges, the average 
slopes at frequent intervals, and, on irregular ground, make 
illustrative sketches to aid in utilizing the other notes. Prac- 
tice gradually teaches how to observe critical points intelli- 
gently, and to record them briefly. In valleys or plains, where 
the location indicated is made up of long tangents and easy 
curves, little detail is required; but on bluffy, tortuous ground, 
with unavoidable divides to overcome, and long reaches of 
maximum gradient to be fitted, the method by contours is not 
only the simplest and clearest way of compiling necessary 
information, but is an aid to the engineer in projecting the 
right line, which no substitute can fully replace. 

5. The writer is forced by the strong constraint of experi- 
ence to differ on this subject with Mr. Trautwine. The dif- 
ference, however, is a permissible one, and implies no 
lack of grateful respect for that veteran engineer, whose 
books are our handy-books, and to whose genius we are all 
debtors. 

6. Having made the map, with ten-foot contours, suppose, 
for example, that a continuous gradient five miles long is to be 
located. Spread the dividers to 500 feet by the scale, start at 
the foot of the ascent, and step up, complying with the general 
trend of the ground, to the summit. This needful preliminary 
gives about the distance you have to work on, which cannot 
in many cases be derived from the experimental line directly. 
The profile furnishes the height to be overcome ; and you are 
thus prepared to assume a summit cut, and determine the 
gradient. Having adopted one, say, of 66 feet per mile, 
observe that this rises five feet in 400 feet. Spread the 
dividers, then, to 400 feet by scale, and stand one leg on or 
near the summit, at a point corresponding to a five or ten unit 
in the elevation of the gradient. That is to say, if the grade 
elevation at the summit be 362, for instance, stand the leg of 
the dividers a little beyond or a little short of the summit, at a 
point where the grade elevation is 365 or 360. Thence, exer 
cising good judgment to conform in a general way to what the 
location ought to be, and to make no angular indirections which 
cannot be closed with the maximum curvature, step forward 
down the incline. Name each step mentally as it is made, 
355, 350, 345, 340, &c., and spot at the same time with a pencil- 
point the contour or half space, directly opposite, correspond' 



THE CURVE -PROTRACTOR. 87 

ing to it in elevation. Connect the pencil-marks with a faint 
dotted line. 

7. Were the ground a straight, regular hillside, the steps 
would be made directly from contour to half-space, thence to 
the next contour below, and the dotted line would mark out a 
tangent conforming exactly to the ground surface. On devious 
slopes, rounding within the limit of the sharpest permissible 
curve, the same exact conformity could be obtained, if desired, 
and a grade-line laid down which should require the least 
possible expense in building. On irregular, winding ground, 
an approximation only to the dotted line can be made: it is 
nevertheless a guide to go by; and, the more nearly the loca- 
tion project approaches it, the lighter will the work of con- 
struction be. The dotted line, in short, is analogous to a 
profile ; and the engineer can prescribe his cuts and fills with 
reference to it, by means of curve or tangent, just as on the 
profile he does the same by means of grade-lines. A fairly 
correct map will enable him to construct a profile from the 
project, and to amend its errors without the trouble and ex- 
pense' of tentative field-work. The writer's habitual practice 
has been to base his preliminary estimates on a profile thus 
deduced from the map; and he recommends the practice to 
others. They will be surprised to observe the likeness between 
such a profile, tolerably well done, and that of the subsequent 
location. 

8. It is a good custom, and one which cannot prudently be 
neglected where long reaches of maximum gradient are en- 
countered, to " slacken grade " on the curves. In making this 
adjustment, the contour map is exceedingly useful. An ap- 
proximate project is first required, in order to determine the 
curvature, and, from that, the varying gradient. The location 
can then be laid down on the map with satisfactory precision. 
Opinions differ as to the right allowance per degree of curva- 
ture, and no experiments seem to have been made from which 
to deduce an authoritative rule. Some say 0.025 per degree 
oer 100 feet; others, 0.05; others, variously between the two. 
Probably 0.05 is the safer rate. This amounts to 2.64 feet on a 
mile of continuous one-degree curve, and makes a nine-degree 
3urve, about the curve of double resistance at ordinary passen- 
.,";er speeds. 

9. In projecting locations, the better way generally is to 
strike the curves first. 



88 



THE CUR VE -PR O TRA CTOR. 



10. The following tables may be of assistance. It was need- 
ful, calculating tliem at all, to calculate them right; but of 
course such exactness as the figures would seem to indicate 
is unattainable in practice. 



|: D d 



\ 



! / 


L ::^^. 


i 




R -- -^ 



11. TABLE SHOWING THE DISTANCE, D, IN FEET, AT WHICH 
A STRAIGHT LINE MUST PASS FROM THE NEAREST 
POINT OF ANY CURVE, STRUCK WITH RADIUS r, IN 
ORDER THAT A TERMINAL, BRANCH HAVING RADIUS 
K = 2 r, AND CONSUMING A GIVEN ANGLE, iC, MAY 
MERGE IN SAID STRAIGHT LINE. 

D = (R — r) X (1 — COS. ic). 



Angle 


Degree op the Main Curve. 




















X. 


2» 


3' 


4° 


6" ■ 


6° 


7" 


8" 


9" 


10» 


D. 


1° 


1.72 


1.15 


0.86 


0.69 


0.57 


0.49 


0.43 


0.38 


0.34 


3° 


4.01 


2.67 


2.00 


1.60 


1.34 


1.15 


1.00 


0.89 


0.80 


4' 


6.88 


4.58 


3.44 


2.75 


2.29 


1.96 


1.72 


1.53 


1.37 


5° 


10.89 


7.29 


5.44 


4.35 


3.63 


3.11 


2.72 


2.42 


2.18 


6° 


15.76 


10.50 


7.88 


6.30 


5.25 


4.50 


3.94 


3.50 


3.15 


7° 


21.49 


14.32 


10.74 


8.59 


7.16 


6.14 


5.37 


4.77 


4.30 


8° 


28.36 


18.91 


14.18 


11.35 


9.45 


8.10 


7.09 


6.30 


5.67 


9° 


35.24 


23.49 


17.62 


14.09 


11.75 


10.07 


8.81 


7.83 


7.05 


10° 


43.55 


29.13 


21.77 


17.42 


14.52 


12.44 


10.89 


9.68 


8.71 



THE C I •/.' VE -PRO TRA C TOR. 



8v^ 



If K = 1| r, use half the tabular distance ; if K = 3 j% use 
twice the tabular distance; if R = 4 r, use three times the 
tabular distance, and so on. 




12. TABLE SHOWI]SrG THE DISTANCE, d, IN FEET, AT WHICH 
CURVES OF CONTKAKY FLEXURE MUST BE PLACED 
ASUNDER IN ORDER THAT THE CONNECTING TANGENT, 
T, MAY BE 300 FEET LONG. 



> 
a. 

D 
O 


- 

Degree of Curve. 


























o 


1° 


2° 


3° 


4° 


5° 


6° 


7" 1 8° 


9' 


10° 


o 






















a; 


d. 


1° 


3.9 


5.24 


5.92 


6.29 


6.35 


6.68 


6.86 


7.00 


7.08 


7.18 


r 


2° 




7 


84 


9.43 


10.38 


11.20 


11.70 


12.20 


12.55 


12.80 


13.06 


2° 1 


3" 








. 


11.77 


13.43 


14.64 


15.68 


16.45 


17.09 


17.61 


18.05 


go 


4° 












15.65 


17.39 


18.76 


19.90 


20.82 


21.64 


22.31 


4° 


5° 








. 






19.54 


21.22 


22.76 


24.01 


25.07 


25.97 


5° 


6° 
















23..32 


25.20 


26.70 


28.00 


29.13 


6° 


7° 








. 






.. 


.. 


27.25 


29.01 


30.58 


31.93 


( 


8° 




















31.05 


32.82 


34.41 


8° 


9° 








. 


.. 


.. 


.. 


.. 






34.82 


36.31 


9° 


10° 
























38.56 


10. 



Examvples. 

A 7° and 4° should be 19.9 feet asunder; a 5° and 9° should 
be 25.07 feet asunder. 

As approximations, for a connecting tangent 400 feet long, 
take twice the tabular distance: for a connecting tangent 200 
feet long, take half the tabular distance. 



90 THE CURVE-PROTRACTOR. 

It is thought by some that the parabola is an ideal curve for 
railroads, and should be adopted; by others, that a spiral or 
parabolic " easement," so called, is sufficient by way of tran- 
sitional flexure from straight line to circular curve; by others, 
that a circular curve compounded with similar terminal curves 
of larger radii is to be preferred; by others, that the circular 
curve alone serves all conditious best. The writer holds with 
the last party up to curves of about 4°, and for sharper ones, 
in the absence of proof to the contrary, believes with the third 
party that circular terminal curves, not less than 200 feet long, 
having half the degree of the main curve, are likeliest to 
meet, in a fair measure, the requirements of actual service. 

Meanwhile the old circular curve continues to do good 
work. On well-regulated lines a curve is usuallj^ indicated to 
the traveller by the inclination of the car only; there is no jar. 
Some years ago one of the most intelligent, experienced, and 
enterprising railroad managements in this country caused a 
thorough practical test to be made of the second device above 
mentioned, with a view to -its adoption if found advisable. 
The engineers and superintendents who made the test reported 
adversely, on experimental grounds. The proposed improve- 
ment was not adopted. 



PROBLEMS m FIELD LOCATIONS. 
XXYI.— XXXVII. 



PROBLEMS IN FIELD LOCATIOISr. 



XXVI. 



HOW TO PROCEED WHEN THE P. C. IS INACCES- 
SIBLE. 

1, Suppose, for example, a pro- 
jected 5° curve, beginning at stake 
24.20, or B in the diagram. 

First Method. — At any point 
A, which we will assume to be 
stake 23. 40, set up the transit. Let 
it be judged that stake 27, marked 
D in the diagram, must fall on ac- 
cessible ground. Then the distance 
B D, around the curve is 280 feet, 
corresponding to an angle E B D of 
7" at the circumference, or an angle 
of 14° at the centre. The chord of 
a 1° curve consuming this angle, by 
Table XVI., is 1,396^6 feet; that of 
a 5° curve, B D in the figure, is one-fifth of this, or 279.3 feet. 
In the triangle A B D are thus known the sides A B, B D, and 
the sum of the angles at A and D, which sum is equal to the 
angle E B D. 

Hence, by trigonometry, — 

As the sum of the sides given = 359.3 A C .... 7.444543 

Is to their difference =199.3 2.299507 

So is tan. h sum of angles at base =3° 30' . . . . S.78G4S0 




To tan. -J their difference 



= 1« 



8.530536 



Adding half the difference to half the sum, the larger angle, 
A, is found to be 5° 26j' ; subtracting half the difference from 
half the sum, the smaller angle, D, is found to be 1° 33^''. The 

93 



94 ffOW TO PROCEED WHEN THE P. C. IS INACCESSIBLE. 



length of the side A D may he found in like manner by trigo. 
nometrical proportion; or, perhaps more simply, thus: — 

B D X nat. cos. D = D F = 279.2. 
B A X nat. cos. A = A F = 79.6. 
AF + FD = AD = 858.8. 

We are now prepared, from our poi]it A, to deflect the angle 
5° 26^' R, and lay out the line A D to the point D on the curve. 
Moving the instrument to that point, and backsighting to A, a 
deflection of 1° 33^' R places the telescope on line DB; a fur- 
ther deflection of 7° places it in tangent at D, and the curve 
may thence be traced in both directions. 

2. Second Method. — Having, as in the first method, 
judged that stake 27, marked D, must fall on accessible ground, 
and thus determined the central angle subtended by the arc 
B D, refer to Table XVI. for the ap. dist. of a 1° curve, corre- 
sponding to 14°, the given angle. It proves to be 703,5 feet. 
One-fifth of this, 140.7 feet, is the tangent or apex distance, 
BC, of a 5° curve, which may be measu'^ed on the ground. 
Moving the instrument to C, turning 14° R, and laying off the 
line C D = B C, the point D on the curve is ascertained. 

3. The preceding methods are manifestly apijlicable to the 
ends also of curves, as well as the beginnings. A case not 

unfrequent in practice may be added in 
conclusion of the subject. 

Suppose a 2° curve terminating at C, in 
marsh or stream not measurable directly. 
Let C fall at stake 32.20. At any con- 
venient point A, say stake 29, place the 
transit with telescope in tangent. The 
arc A C, = 320 feet, includes an angle of 
go 24'. The ap. dist. of a 1° curve corre- 
sponding to this angle in Table XVI. is 
320.34 feet; that of a 2° curve is therefore 
1G0.2 = A B. Move to B, deflect 6° 24^ li 
into the range of the terminal tangent, 
and fix E on the opposite shore. Fix also 
D, and note the angle EBD. Move to 
E. Measure the angle DEB, and the distance D E. The tri- 
angle BED may then be solved. If B E is found to be 670 
feet, C E = 670 — 160.2 = 509.8, and stake E = 32.20 + 509.8, 
— say 37.30. 




now To PRCCEED WHEN THE P. C. C. IS INACCESSIBLE. 



XXVII. 




HOW TO PROCEED WHEN THE P. C. C. IS INAC- 
CESSIBLE. 

1. Suppose a 4° curve, 
A B, compounding at B into 
a 6° curve B C. 

FiKST Method. — Place 
the transit at any point A, 
say stalce 34. Let the pro- 
posed P. C. C. fall at stake 
3G.25. Assume that we wish 
to reach C, on the second 
curve, by means of the 
straight line ADC. The 
arc AB, covering 225 feet 
of a 4° curve, subtends an 
angle of 9°. A D is half the chord of twice this angle. 

By Table XYI., the chord of 18° on a 1° curve is 1,792.7 feet. 
That of a 4° curve is therefore 448.2 feet, half of which = 
234.1, = A D. The mid. ord. of 18° on a 1° curve, by the same 
table, is 70.54 feet; one-fourth of which, or 17.635, is the mid. 
ord. B D, corresponding to the same angle on a 4° curve. In 
order to find what angle on the 6° curve this mid. ord, B D, — 
17.635 feet, corresponds to, multiply it by 6, and seek the prod- 
uct, 105.81, in Table XVI., where it is found, nearly enough 
for field-practice, opposite the angle 22^ 04'. The chord of that 
angle, on a 1° curve, is seen at the same time in the adjoining 
column to be 2,193.2 feet; on a 6° curve it is therefore 365.5 
feet, one-half of which, = 182.75 feet, = DC, and one-half of 
21° 04' = 11° 02', = the angle covered by the arc B C. Thus 
are found the angle at A = 9°, the angle at C = 11° 02^, and 
the distance AC = 224.1 + 182.75, = 406.85 feet. The angle 
11° 02' corresponds to a length of 1.84 feet on the 6° curve; 
C, therefore, falls at stake 36.25 + 1.84 = 38.09. With these 
data the field-work is obvious. 

2. Second Method. — Having reached the point A, and 
determined the arc A B = 9°, as above, find in Table XVI. the 
ap. dist. 450.95 feet, corresponding to the given arc, one-fourtb 



96 



TO SHIFT A P. G. 



of which = 112.7 feet, = ap. dist. for the 4° curve. Move to 
E, deflect 9° R; range out the line E F, made up of E B = A E 
^ 112.7 feet, and BF any convenient distance, say 90 feet. 
Tills 90 feet is the assumed ap. dist. of some unknown angle on 
Ibe 6° curve. To find the angle, multiply 90 by G, and seek the 
product, 540, in the AD column of Table XVI., where it is 
found opposite 10° 46'. By moving then to F, deflecting 10° 
46' R, and measuring F C = 90 feet, the point C is fixed on the 
second curve. 

3. Should unexpected obstacles be met in carrying out either 
of these plans, the triangles AGC or EGF may be solved, 
and the point C fixed by means of the lines AG, G C. 

4. The application of the foregoing methods to .-urning 
obstacles on simple curves needs no special instance. 



XXVIII. 

TO SHIFT A r. C. SO THAT THE CURVE SHALL 
TERMINATE IN A GIVEN TANGENT. 




1. Suppose a 3° curve AB to 
have been located, containing an 
angle of 44° 26', and ending in 
tangent B E : required, that it 
shall end in tangent D F, parallel 
to B E. It is plain, from the 
diagram, that if the curve and 
its initial tangent be moved forward, like the blade of a skate, 
until the terminal tangent merges in D F, the P. T. will hr.ve 
traversed the line B D, equal and parallel to A C. If, there- 
fore, on the ground at B, the angle E B D, equal to the whole 
angle consumed by the curve, in this case 44° 26', be laid off 
to the right, and the distance B D to the range of the proposed 
terminal tangent be measured, the equal distance AC, from 
Ibe original to the required P. C, is thus directly ascertained. 
Should such direct measurement be impracticable, range out 
the tangent BE, and, at any convenient point, measure the 
distance from it square across to the proposed terminal tan- 
gent D F, say 56 feet. Then in the right triangle BED, mak- 
ing BD radius^ we have given the angle at B = 44° 2&j and 



TO SUBSTITUTE A CURVE OF DIFFERENT RADIUS. 97 

the sine E D = 56 feel. Hence, by trigonometry, E D -^ nat 
sin. 440 26', or 56 -^- 0.7, = B D = 80 feet, = distance A C along 
the initial tangent, from the erroneous to the correct P. C. 

2. This i^roblem occurs more frequently than any other in 
the field; and the young engineer should have it by heart, that 
the distance square across between terminal tangents, divided 
by the natural sine of the total angle turned, will give him the 
distance he is to advance or recede with his P. C. to make a fit. 

3. Excepting on precarious rocky steeps, city streets, or like 
exact confines, to strike within two feet of any point desig- 
nated in the project, may be considered striking the mark. 
Astronomical nicety, whether with transit or level, in an ordi- 
nary railroad location, is mere waste of time. 

4. The observant reader will not fail to perceive that the 
foregoing rule applies to systems of curves, or to compound 
lines also, the angle E B D being the angle included between 
the initial and terminal tangents, let what flexures or indirec- 
tions soever have been interposed; and that, if the angle re- 
ferred to be either 180° or 360°, adjustment by shift of P. C. is 
impracticable. In those cases, a change of radius becomes 
necessary. 



XXIX. 

TO SUBSTITUTE FOR A CURVE ALREADY LOCATED, 
ONE OF DIFFERENT RADIUS, BEGINNING AT THE 
SAME POINT, CONTAINING THE SAME ANGLE, AND 
ENDING IN A FIXED TERMINAL TANGENT. 

1. Suppose the 4° curve AB, 
containing an angle of 32° 20', to 
have been located, and that it is 
required to substitute for it an- 
other curve A C, which shall end 
in a parallel tangent C F, 60 feet 
to the right. 

First M k t h o d. — Find the 
length of the long chord A C, = 

AB + B C. Referring to Table XVL, the chord of a 1° curve 
for 32° 20' is seen to be 3,190.8 feet; that of a 4° cui've, there- 




98 TO FIND THE POINT AT WHICH TO COMPOUND. 

fore, = 797.7 feet, say 798 feet, = AB. To find BC, solve 
the triangle B D C, observing that the angle DBC = BAI = 
one-half of the central angle 32° 20', = 16° 10', and that D C 
= 60 feet. Then D C -f- nat. sin. 16° 10' = 60 -i- .278 = say 
216 feet, = B C. Hence AC = AB + B C = 798 + 216 = 
1,014 feet. 

Having thus found the length of chord A C, the radius and 
rate of curvature may be deduced as in X. 

Or, dividing the tabular chord of 32° 20' by chord A C = 
1,014, the degree of the required curve is ascertained directly 
to he 3.15, equivalent to 3° 09'. 

2. Second Method. — Find the apex distance AH, = AI 
-f I H. The tabular ap. dist. of 32° 20' divided by 4 gives A 1 
= 415 feet. In the triangle KDC, the side DC -^ nat. sin. 
K = 60 -^ nat. sin. 32° 20', = 112 feet = KC = I H. Then 
AH = AI 4- IH = 415 + 112 = 527 feet; and the tabular 
ap. dist. 1,661 -^ 527 gives 3.15, equivalent to 3° 09', the degree 
of the required curve A C, as before. 



XXX. 

HAVING LOCATED A CURVE A B C, TO FIND THE 
POINT r, AT WHICH TO COMPOUND INTO ANOTHER 
CURVE OF GIVEN RADIUS, WHICH SHALL END IN 
TANGENT E F, PARALLEL TO THE TERMINAL 
TANGENT OF THE ORIGINAL CURVE, AND A GIVEN 
DISTANCE FROM IT. 

1. To find B, the angle B I C must be 
found. Call the given distance between 
tangents D; the larger radius, R; the 
smaller one, r; the required angle, a 
Then, referring to the figure, observe 
that in the triangle I H K, I H being ra- 
dius, IK is the cosine a; i.e., IK -^ IH 
= nat. cosine a. But I H = R — r; IK 
= IC — KC, and KC = KF or HE 

+ FC, =r + D; i.e.,IK = Il — r — D. Hence naU cosine 

(t = (R - 7- - D) -r- (R -- r) = 1 ~ D H- (R - ?•). 




TO SHIFT A P. C. C, 



99 



The same reasoning would apply if A B E were the curve 
first located, and a terminal curve of larger radius required to 
be put in. 

2, We have, then, the following general rule for such cases: 
Divide the perpendicular distance between terminal tangents 
by the diffei-ence of the radii, and subtract the quotient from 
unityi the remainder is the natural cosine of the angle of re- 
treat along the located curve to the required P. C. C. 

Example. 

3. A 3° curve on the ground, to find the P. C. C. of a 5° curve 
striking 27 feet to the right. Here D = 27; R — ? = 1,910 — 
1,146, = 764; D ^(R - r) = 27 ^ 734, = .03534; and 1 — 
.03534 = .96466 = nat. cosine 15° 17'. We must go back, 
therefore, .509 feet on the 3° curve, to compound into the 5° 
curve. Had the 5° curve been located first, we must have 
gone back 306 feet to begin the 3° curve which should strike 27 
feet to the left. In either case, time might be saved by moving 
directly from E to C, or the reverse, and spotting in the curve 
backwards. To do this, we have in the right triangle F E C, 
the angle E = half of 15° 17', = 7° SSi', and the side F C = 27 
feet. Then E C = 27 -^ nat. sin. 7° 38^', = 203 feet; and if E 
were stake 54.20 on the 5° curve, B would fall at stake 54.20 
— 3.06, = 51.14; and C, the P. T. of the 3° curve, at 51.14 -j- 
5.09, = stake 56.23. 



XXXI. 



TO SHIFT A P. C. C, SO THAT THE TEPvMINAL 
BRANCH OF THE CURVE SHALL END IN A 
GIVEN TANGENT. 

First Case: the terminal branch 
having the shorter radius. 

1. Suppose the compound curve 
ACN located, and that it is required 
to fix a new P. C. C. at B, from which 
the terminal branch BM shall merge 
in tangent ML, a given distance from 
NO. To fix B, the central angle 
B H ]M of the new terminal branch 
must be found, and substituted for 
Cl^, C^U tii§ iQUgyr r^^m lij the sUorter Qiie, r/ the 4i§' 
L.DfC. 




100 



TO SHIFT A P. a C. 



tance asunder of the terminal tangents, D ; the central angle, 
C I N, = I E K, of the located terminal hranch, b ; and the 
central angle, B H M, = HE F, to be substituted for it, a. 

In the right triangle, EIK, EK = EI cos. I E K = (R — r) 
COS. b. 

In the right triangle HFE, EF = EH cos. H E F = (R — r) 
COS. a. 

Also, F K = L O = D, since each is equal to r — K L. 

ThenEF = EK — FK; i.e., (R — r) cos. a=(R — r) cos.b 
— D. 

Hence nat. cosine a = nat. cosine b — [D -^ (R — r)\. 

Were the curve B M- located, and the curve C IST to be substi- 
tuted for it, — that is to say, were a given and b required, — 
we should have, by transposition, nat. cos. b = nat. cos. a -j- 
D -f- (R - r). 

Example. 

A 3°, compounding into a 5° curve at C, which consumes an 
angle CIN, ;= 30° 22', and ends in a- tangent, NO, which is 
iound, by measurement of L O, to be 34 feet too far to the left. 

Here, D = 34, R = 1,010, r = 1,146, b = 30° 22'; and, by 
the solution, nat. cos. a = nat. cos. 30° 22' — 84 -r- [1,910 — 
1,146] = 0.8628 — (84 - 764). 



34 

764 

.0445 



log. 1.531479 
log. 2.883093 



loir. 2.G4S386 



Then 0.8628 — 0.0445 




with central angle I E F 



0.8183 = cos. 35° 05', = angle a ; 
a — 6 = B IT M — CIN = B E C 
= the angle of retreat from the 
erroneous P. C. C. = 35° 05' — 
30° 22' = 4° 43', equivalent to b":? 
feet, on the 3° curve, from C to B. 

2. Second Case: the terminal 
branch having the longer radius. 

Let B K represent the terminal 
branch located with central angle 
I K O = 6, and suppose it required 
to determine the new arc CM, 
a. Call the longer radius' R, the 



shorter one r; the distance L N between tangents, D. In the 



TO SHIFT A P. C. C. 101 

right triangle IKO, KO =KI cos. IKO =(R — r) cos. b. 
Ill the right triangle FIE, EF = EI cos. lEF = (K — r j 
COS. a. Also, E H = L N = D, since each is equal to R — K L. 

Then EF = EII + IIF = EH + KO; i.e., (R — r) cos. 
a={U — r) COS. 6 -j- D. Hence nat. cos. a = nat. cos. 6 + 
D -- (R - r). 

Were the curve CM located, and the curve BN to be sub- 
stituted for it, that is to say, were a given and b required, 
Ave should have, by transposition, nat. cos. b = nat. cos. a — 

D - (R - r). 

Example. 

A 5° compounding into a 3° curve at B, which consume? 
{■-n angle of 44° 20', and terminates at N, 28 feet too far to 
t::o left. Here D = 28, R = 1,1)10, r = 1,146, b = 44° 20; 
and, by the solution, nat. cos. a = nat. cos. 44° 20' + 28 
H- 764. The nat. cos. 44° 20' = 0.71529 ; 28 -=- 764 = log. 
1.447158 — /oj. 2.883093 = Zof/. 2.i:04G65, corresponding to the 
decimal 0.003C5, which, being added to nat. cos. 44° 20', gives 
(;.75194, the nat. cos. 41° lo . Then B K ?T — C E M = 44° 20' 
— 41-15' = 3° 05'= angle BIC, equivalent on a 5° curve to 
62 feet, which therefore is the distance around the arc from 
B, the erroneous P.C.C., to C, the correct one. 

3. Fro^n these formulas the follovving general rule may be 
drawn: Divide the distance between terminal tangents by the 
difference of the radii, and call the quotient Q. Find the nat- 
ural cosine of the terminal arc already located, and call it C. 
The sum or the difference of Q and C will be the natural 
cosine of the terminal arc to be substituted for that already 
located. With radii in the order R, r, should the terminal 
inside 



tangent located strike s f -a \ *^® proposed tangent; or, 
with radii in tlie order r, R, should the terminal tangent 
located strike ] • -i [ the proposed tangent, — take the 

Trc > of Q and C for the required cosine. 

^ difference ) ^ 



102 



TO FIND THE POINT AT WHICH TO BEGIN A CURVE. 



XXXII. 



HAVING LOCATED A TANGENT, A B, INTERSECTING 
A CURVE, C D, FROM THE CONCAVE SIDE, TO 
FIND THE POINT E ON SAID CURVE AT WHICH 
TO BEGIN A CURVE OF GIVEN RADIUS WHICH 
SHALL MERGE IN THE LOCATED TANGENT. 

/' 1. Place the transit at the 

intersection point B. Set 
points at equal distances 
therefrom in both directions 
on the curve already located, 
by means of which the direc- 
tion of a tangent to that curve 
at B may be fixed, and the 
angle F B A measured. Call 
that angle a ; and, as shown 
in the figure, suppose the lo- 
cated curve to be prolonged in- 
to a terminal tangent, parallel 
with the newly located tan- 
gent A B. Complete the dia- 
gram. Call the larger radius R; the proposed radius, r; the 
central angle of the proposed curve, x. Then, obviously, the 
line A G = II cos. a. It is also equal to (R — r) cos. x -\- r. 
That is to say, R cos. a = (R — r) cos. x -}- r. Hence cos. x 
== {"Rcos. a — r) -i- (R — r); and x — a = angle BGE, sub- 
tended by the arc B E, from which the length of the arc may 
be deduced, and the point E ascertained. 

Example. 
DC, a 1° curve; angle a = 64° 32': to connect with a 49 
curve. Here cos. x = (5,730 X 0.43 - 1,433) -5- (5,730 - 1,433) 
= 0.24 = cos. 76° 06'; and x — a = ll° 34', equivalent to a dis- 
tance from B around the 1° curve of 1,157 feet to E, the point 
at which to begin the 4° curve. 




TO LOCATE A Y. 



103 



XXXIII. 



HAVING LOCATED A TANGENT, A B, INTERSECTING 
A CURVE, C J), FROM THE CONVEX SIDE, TO FIND 
THE POINT K ON SAID CURVE AT WHICH TO 
BEGIN A CURVE OF GIVEN RADIUS WHICH 
SHALL MERGE IN THE LOCATED TANGENT. 

L This pr()l)lem is analo- 
gous to the preceding one. 
The preparatory steps are the 
same in both. Having found 
the angle «, however, it will 
be manifest to the attentive 
reader, that, in this case, R 
coH. rt = (K -j- r) COS. X -{- r. 
Hence cos. a: = (R cos. a — r) 

-^ (R + r). 

Example. 
2. D C, a 1° curve; angle a = 64° 32': to connect with a 4° 
curve. Here cos. x = (5,730 X 0.43 - 1.433) ^ (5,730 + 1,433) 
= 0.1439 = COS. 81° 43'; and x — a = 17° il', equivalent to a 
distance from 13 around the 1° curve of 1,718 feet to E, the 
point at which to start tlie 4° curve. 




XXXIV. 



TO LOCATE A Y. 



B 


D 


A 






\ ^^^ 


.-^ 


























/ ^ 


\ \ / "^ 


N 






/ 




\ 










\ 


c 


/ 




"N 


^' 


\ 




^'"'^'x'^ 




\ 


'\.^ 


\c 




\ 


^--^^ 






{ 


3 









tersecting ilie tangent BA. 



1. The processes of the 
two former problems may- 
be adopted. In this case 
the angle a vanishes, and 
the COS. X clearly is equal to 
(R-r)-^(R + r). 

2. Another solution of tlie 
Y problem is as follows : — 

Draw tlie tangent E D in- 
Then is BD = D A, for the lea- 



104 



TO LOCATE A Y. 



son that each is equal to DE. Make GF = R + r, the diame- 
ter of a semicircle. Said semicircle touches tangent B A at D, 
its middle point; and D E being perpendicular to G F, we have 
by geometry GE : DE ;: DE : EF; i.e., GE X EF, orR X 
r, = DE2. Hence DE = B D = D A = V^R X r = R tan. i a-, 
and we are thus enabled to fix the points E and A. 

3. In the two foregoing problems, the angle consumed by 
curve E A is = 180° — x. 

Example. 

BE, a 21° curve located; BA, a tangent: to complete the 
Y with a 6° curve, E A. 

By the first method, cos. x = (R — r) -^ (R + r) = (2,292 
— 955) -^ (2,292 -j- 955) = 1,337 ^ 3,247 = log. 3.126131 — 
log. 3.511482 == 1.614649, which corresponds to log. cos. 
9.614649, or to the decimal number 0.4118, indicating in either 
case the angle 65° 41' = x. 

DE = BD = DA = R tan. ix = 2,292 X 0.6455 = 1,479.4. 
DE may be found also by reference to Table XYL, where the 
ap. dist. of a 1° curve for 65° 41' is seen to be 3,698.6. Dividing 
this number by 2|, we have 1,479.4, as above. 

Or, by the second method, — 

D E = VR X r = \/2188860= 1,479.4. 

Having thus the means 
of fixing points E, D, and 
A, the curve E A can be 
laid down. 

4. If B A is curved con- 
vex to the Y, construct 
the figure as in margin, 
and reason thus: — 

In the triangle EGF, 
formed by lines connect- 
ing the curve-centres, the 
sides are respectively 
equal to the sums of the 
contiguous radii : the 
angles may therefore be 
found as in Case III., 
Trigonometry. 
Lines drawn bisecting the central angles of the several 




TO LOCATE A Y. 105 

curves will pass throiigli the points of intersection of the tan- 
gents to those curves severally. But lines so drawn in this 
case bisect also the angles of a triangle, and, demonstrably 
by geometry, meet in one point equidistant from the three 
sides of the triangle. That point, therefore, must be a com- 
mon P. I. for all the curves, and that equidistance the "ap. 
dist." length common to them all. 

Example. 
Given B A, a 3°, and B C, a 4° curve : to complete the Y with 
a 5° curve, C A. 

E F = 1,910 + 1,146 = 3,056. 
GF = 1,433 + 1,146 = 2,579. 
E G = 1,910 + 1,433 = 3,343. 

Then, by Case III., Trigonometry, — 

As EG, 3,343. . . . log. (a. c.) 6.475864 

Is to E F + G F, 5,635 .... log. . . 3.750894 
SoisEF— GF, 477 . . . . log. . . 2.678518 

To diff. of segments of E G,"804 ...'.. 2.905276 

Adding half the difference to half the sum of the segments 
of the base EG, we shall have the greater of them; i.e., 
(3,343 -f- 804) -i- 2 = 2,073.5, which is the cos. E, E F being 
radius. Hence 2,073.5 H- 3,056 = log. 3.316704 — log. 
3.485153 = 9.831551 = cos. 47° 16' = E. By Table XVI., the 
ap. dist. of a 1° curve corresponding to this angle is 2,507 3 : 
that of a 3° curve, therefore, is 835.8 = the common ap dist. 
B D or D A. Multiplying the common ap. dist. by 4, we shall 
find opposite the product in Table XVI. the central angle of 
the 4° curve to be 60° 32'; multiplying it by 5, we find, in like 
manner, the central angle of the 5° curve to be 72° 12'. Arc 
B A, = 47° 16', is equivalent to 1,575 feet on the 3° curve; arc 
BC, = 60° 32', is equivalent to 1,513 feet on the 4° curve. 
Points being thus fixed at A and C, curve C A can be laid on 
the ground. 

5. If curve B A is concave to the Y, the radii being given, 
construct the figure as follows: — 

First draw the triangle GFE, the sides of which are obvi- 
ously derived from the given radii. Prolong the sides E G and 
E F indefinitely. Bisect the exterior angles at G and F with 



106 



TO LOCATE A Y 



lilies meeting at I), and from D let fall perpendicnlars on E H. 
EA, and G F. Then, comparing triangles GBD, GCl), IIif 
angles at G are equal by construction ; the angles at B and C 
are right angles, the side G D common. Hence the triangles 
are equal in all their parts: B G = G C, and B D = D C. By 
like reasoning, it appears that C F = F A, and DA = DC. 
The point D being equidistant from the right lines E B, E A, 
which limit angle E, a line bisecting that angle will strike 
point D. 




6. It may be remarked, therefore, that lines bisecting the 
vertical angle and the exterior angles contained between the 
base and the prolongation of the sides of any triangle, will 
meet in a point equidistant from the base and the said prolon- 
gations. We thus have in the figure all the conditions for fit- 
ness of tlie curves. It remains only to solve the triangle G F E, 
seeing that from its angles the required central angles can be 
obtained. 

Example. 

B A, a 1°, BC, a 6° curve, located: to complete the Y with 
an 3° curve, C A, 



TO LOCATE A TANGENT TO A CURVE. 107 

In triangle G F E, — 

E F = 5,780 — 717 = 5,013. 
E G = 5,730 — 955 = 4,775. 
G F = 955 + 717 = 1,672. 

Then, by Case III., Trigonometry, — 

As E F . . . 5,013 .... log. (a. c. ) 6.299902 
Is to E G + G F, 6,447 .... log. . . 3.809358 
SoisEG — GF, 3,103 . . . . log. . . 3.491782 

To diff. seg. of base, 3,991 . . . log. . . 3.601042 

The longer segment, therefore, is 4,502; the shorter, 511- 
Cos. E ^ the longer segment divided by E G = 4,502 -^ 4,775 = 
lo[i. 3.65.3405 — 3.678973 = 9 974432 = cos. 19° 28' = angle E. 

Cos. GFE = the shorter segment divided byGF = 511-^ 
1,672 = log. 2.708421 — log. 3.223236 = 9.485185 = cos. I'l^ 
12/ = angle GFE. 

The central angle, B G C, of the 6° curve, is equal to 180 — 
F G E = the sum of the angles at E and F = 72° 12' + 19° 
28' = 91° 40', making the arc B C = 1,528 feet. The arc B A, 
equivalent to 19° 28' of a 1° curve, = 1,947 feet. Points C and 
A being thus ascertained, curve AC maybe located. It will 
consume an angle = 180° — 72° 12' = 107° 48', equivalent, on 
an 8° curve, to 1,347.5 feet. 



XXXV. 

10 LOCATE A TANGENT TO A CURVE FROM AN 
OUTSIDE FIXED POINT. 

1. If the ground is open, and the curve can be seen from the 
fixed point, it may be marked by stakes or poles at short inter- 
vals, and the tangent laid off without more ado. 

2. Suppose, however, that on cumbered ground a trial tan- 
gent, A B, has been run out, intersecting the curve at B: it is 
required then to timl the angle BAE, \\\ order that the true 
tangeut.AK may be laid down. 



108 



TO SUBSTITUTE A CURVE. 



Example. 

A B = 1,500 feet ; D H B, a 4° curve; angle F B D = 20° 13'. 

First, the angle FBD, between a tangent and a chord, is 
^(jiial to half the central angle subtended by the same chord. 
Angle D C B, therefore, = 40° 26'. By Table XVI., the chord 
of 40° 26', for a 1° curve, = 3,960.2 feet; for a 4° curve, it is, 
say, 990 feet = D B; and D I = I B = 495 feet. The mid. ord. 
H I is, in like manner, found to be 88.25 feet. Deducting this 
from the radius of the 4° curve, we have I C = 1,344.4 feet. 

Then lC-^lA = tan. I AC; i.e., 1,344.4 H- (495 + 1,500) 
= 0.674 = tan. 33° 59' = angle I AC. 




Next, by geometry, the proposed tangent A E =\/A D X A B 
= V2,490 X 1,500 = 1,932.6 ; and E C -^ A E = tan. E A C = 
1,432.69^ 1,932.6 = 0.7413 = «an. 36° 33' = angle E A C. Then 
E A C — I A C = .36° 33' —33° 59' =2° 34' = angle B A E, the 
angle required, which can accordingly be laid off from the fixed 
point A, and the tangent located. 



XXXVI. 



TO SUBSTITUTE A CURVE OF GIVEN EADIUS FOR 
A TANGENT CONNECTING TWO CURVES. 

Example. 

1. A B, a 4° curve; BC = 774 feet; CD, a 6° curve: to put 
in the 1° curve, EF. 

Sketch the figure as in margni, HK being parallel and equal 
lo BC. Then KG = BG — BKorCH = 1,433 — 955 = 
478 feet; KH -f- GK = 774 -^ 478 = 1.62 = tan. 58° 19' = 
angle KGH; and KH -^ ain. 58° 19' =774 4-0,851 = 909,6 
ieet = GH. 



TO RUN A TANGENT TO TWO CURVES. 



109 



In the triangle GHI we have then the sides given; namely 
GH = say, 910 feet, HI = 5,730 — 955 = 4,775 feet, and GI 
= 5,730 — 1,433 = 4,297 feet: to 
liud the angles. 

Under Case 3, Trigonometry 

(III.), IH : IG + GH :: IG 

— GH:IL — LH; i.e., 4,775 : 
5,207 : : 3,387 : 3,093, the differ- 
ence of the segments into which 
the base I H is divided by a 
perpendicular from G. Adding 
half the difference of the seg- 
ments thus found to half their sum, the longer segment, I L, is 
found to be 4,234 feet ; subtracting half the difference from 
half the sum, the shorter segment, L H, is found to be 541 feet. 
Then H L -i- H G = 541 ^ 910 = 0.5945 = cos. 53° 31' = 
angle GHI. In like manner, dividing I L by I G, we find the 
angle GIH to be 9° 40'. The sum of these angles = angle 
E G H = 63° 20', for the reason that each is equal to 180 — 




II G I. Finally, E G H — K G H = 63° 20' — 58° 19' 



5° 01' 



= angle E G B, equivalent to a distance from B of 125 feet 
around the 4° curve to the P. C. C. at E; and GIH — EGB 
= 9° 49' — 5° 01' = 4° 48' = angle CHF, equivalent to a 
distance from C of 80 feet around the 6° curve to the P. C. C. 
atF. 



XXXVII. 

^O RUN A TANGENT TO TWO CURVES ALREADY 
LOCATED. 




then be 
the line 



1. If one curve be visible 
from the other, or if both 
be visible from some inter- 
mediate point, ■ mark them 
on the ground with stakes 
at short intervals. The 
points M or L in the range 
^ of the required tangent may 

fixed by one or two trial settings of the transit, and 
put in. 



110 TO RUN A TANGENT TO TWO CURVES. 

2. Should obstacles prohibit this plan, measure any ccn- 
venient line, FG or B CD, from one to the other curve, and, 
completing the traverse AFGrE or ABODE, determine 
thence the bearings and distances asunder of the centres A 
and E. The right triangle A E K, in which E K = the sum of 
the radii, may then be solved, and the points H and I ascer- 
tained as in the following example : — 

Example. 

F B, a 4° curve ; G D, a 6° curve. N. S. E. W. 

A B, N. 20° E., 1,433 feet . . 1,346.6 490.0 

BC,East, 3,570 feet . . - 3,570.0 

C D, N. 34° E., 1,800 feet . . 1,492.2 1,006.2 

955 feet . . 675.2 - 675.2 



3,514.0 5,066.2 675.2 



Then 4,391 -f- 3,514 = 1.2496 = tan. 51° 20' = bearing 
AE; and 4,391 -^ sin. 51° 20' = 5,624 feet = distance A E. 
Also, EK -f- AE = (1,433 + 955) ^ 5,624 = .sm. 25° 08' = 
angle EAK; and angle AEK = 90°00'— 25°08/= 64° 52'. 
Hence the bearing of A K or HI is N". 76°28/E., and that of 
A H or I E, K 13° 32^ W. 

Since AB bears X. 20° E., the angle HAB = 20° 00' + 13^ 
^2'^::^ 33° 32', equivalent to a distance of 838 feet from B around 
the 4° curve to the required P. T. at H; and, since DE benrs 
N. 45° 00' W., the angle IE D = 45° 00' — 13° 32/ = 31° 28 . 
equivalent to a distance of 524 feet from D around the 6° curve 
to the required P. C. at I. 

3. Should the curves turn in the same direction, the side 
EK of the triangle AEK is equal to the difference of the 
radii instead of their sum. In other respects, the method 
exemplified will apply to that case also. 

4. The preceding solution may be useful as an exercise. 
But the problem is one of rare occurrence, and the conditions 
must be extraordinary which prevent a close approximation, 
at least, to the true line in the field. The better way in actual 
l)ractice, then, is to run out a trial tangent as nearly right as 
j.ossible. If it errs by passing outside the objective curve, 
ciose with a compound (XXIX.) ; if that error be inadmissible, 
or if it ei-rs by cutting the objective curve, measure the miss, 
and divide it by the length of the trial tangent. The quotient 



TO RUN A TANahNT TU TWO CURVES. \\\ 

will be the iiatural tangent of the angle of retreat or advance 
on the first curve recpiired to make the tangent fit. 

5. A still closer ad.iustnien', would be, after determining the 
angle approximately as above, to find the "tangents" corre- 
sponding to it for the two curves in Table XYI. Subtract the 
sum of these tangents from the length of the trial line, if it 
cuts the objective curve; add the sum, if it passes outside. 
AVitli the number thus found, divide the measured amount of 
error for the tangent of the angle of retreat or advance, as the 
case may be. 

G. Suppose, for illustration, that a trial tangent, bearing by 
needle X. 54° 30' E., is run out from stake 24.80 of a 4° curve, 
intending to touch a G°, but is found to cut it. Supjwse fur- 
ther that the objective 0° curve was laid down and numbered 
in the direction of approach towards the 4° curve; that its P. 
C. is stake 25,10, and the magnetic bearing of its initial tan- 
gent S. 30° 30' W. The angle, then, between the bearing of the 
trial tangent and tliat of the initial tangent of the G° curve, is 
24°, corresponding to a distance of 400 feet on the latter curve. 
At stake 25.10 + 4.0 = 29.10, therefore, a tangent to the G° 
curve would oe parallel to the trial tangent. Go forward on 
the trial tangent, accordingly, to a point oi^posite 29.10, and 
measure the distance square across to that plus on the G° 
curve. Assuming the trial tangent to be 2,500 feet long, and 
the amount of the miss to be 87 feet, the nat. tan. of the 
angle of error is 0.0348 = tan. 2°. By the method in (4), this 
calls for a shift of the P. T. 50 feet ahead on the 4° curve, 
making the new P. T. 24.80 + 0.50 = stake 25.30, and ad- 
vances the P. T. of the 6° curve to stake 29.43 of that numera- 
tion. 

The method in (5), applied to this case, brings the angle of 
error 2° 02', instead of 2°, equivalent to a deviation of 1^ feet 
scant in half a mile from the line corrected by the method in 
(4), and agreeing exactly with the correction determined by 
the method in (2). 



TRACK PROBLEMS 
XXXYIII.— LI. 

(Standard Gauge 4 Feet 8i Inches.) 



TRACK PROBLEMS. 



XXXVIII. 

REVERSED CURVES. 
The following problems will be useful in laying off turnouts, 
the adjustment of tracks near stations or shops, and the like; 
but reversed curves should never be used on the main line 
between stations, where they are both objectionable and unne- 
cessary. Ground which allows any permissible location 9t all 
will allow straight reaches of at least two hundred or three 
hundred feet between curves of contrary flexure; and in every 
case it is worth the small additional outlay to make such a 
location. 



XXXIX. 



TO CONNECT TWO PARALLEL TANGENTS BY A 
REVERSED CURVE HAVING EQUAL RADII. 

1. The radius R, and the perpendicular distance D, between 
the tangents given. j^c 

1 




115 



116 TO CONNECT TWO PARALLEL TANGENTS. 

Draw the tangents, radii, and curves, fixing the P. R. C. 
midway of D. 

Draw the chords G I, I E, the line B F perpendicular to G I, 
and the line E H in prolongation of radius C E to an intersec- 
tion with B H passed through centre B parallel to tangents. 

That I falls midway of D, follows from the necessary sym- 
metry of the figure ; and G I E must be a straight line, because 
the radii B I, I C, perpendicular to a common tangent at the 
same point, form a straight line, to which the chords G I, IE, 
are equally inclined. 

C H -^ C B = COS. A; but C H = 2 R — D, and C B = 2 R. 

.-. COS. A = (2R — D) -^2R. 

BH = B C sin. A = 2 R sin. A; 

GF = R sin. i A; GE = 4 GF. 

.-. GE = 4 R sin. i A, and GI or IE = 2 R sin. i A. 

Observe, that, in the right triangles G K E and B G F, the 
angles at G and B are each equal to ^ A : hence the triangles 
are similar. 

Example. 

K = 800 feet, D = 24 feet. 
To find angle A. 

Cos. A = (2 R — D) -^ 2 R = 1,576 ~ 1,600 = 0.985 = nat 

COS. 9° 56^ 

BH may then be found = 2 R sin. A = 1,600 X 0.1725 = 
276 feet, and laid off from the P. C. at G to K, the point E 
being fixed by a right angle from K. 

Or GE may be found = 4 R sin. i A = 3,200 X 0.866 = 
277.1 feet, and laid off from G to E, the point I being fixed 
138.5 feet from G, and angle KGE made equal to half of A = 
4° 58'. 

2. The distances G K and D given, to find R. 

In triangle G K E, K E = D. 

D ^ GK = tan. i A; D -4- sin. i A = GE; and GE -f- 
sin. I A = 4 R. 

Or, having found GE, we have from the congruity of trian- 
gles GKE,BFG, 

D : GE :: ^GEorGF : R. 
.-. R = GE2-f-4D. 



TO CONNECT TWd PARALLEL TANGENTS. 



in 



Example. 
GK = 300 feet, D = 28 feet. 

D-^GK Log. 28 ... . 1.447158 

Log. 300 ... . 2.477121 

= Tan. ^ A . 5° 20' . . . . 8.970037 

D -^ sin. i A . . . . Log. 28 ... . 1.447158 

Sin. 5° 20' . . . . 8.968249 

= G E . . 301.24 . . . . 2.478909 

GE-^sm. iA . . .Sm. 502O' . . . .8.968249 

= 4 R . . 3,241 .... 3.510660 
.-. 11 = 810.2. 



XL. 



TO CONNECT TWO PARALLEL TANGENTS BY A 
EEYERSED CURVE HAYING UNEQUAL RADII. 




1. Given the perpendicular distance, D, between two paraL 
lei tangents, and the unequal radii, R and r, of a reversed 
curve, to find the central angles. A, the chords, and the 
straight reach, G K, of the curve. 



118 TO CONNECT TWO PARALLEL TANGENTS. 

Cos. A = C H ^ B C; but C H = (R + r) — D, and 
B C = R + r. 

.-. Cos. A = (R + r — D) -^ (R + r). 

The straight reach GK = BH=(R + r) sin. A. 
The sum of the chords G E = G K -h- cos. ^ A. 

G I = 2 R sin. i A. 
IE = 2 r sin. i A = GE — GI. 

Example. 
D = 28, R = 955, r = 574. 

Cos. A = (R 4- r — D) — (R + r) = 1,501 -^ 1,529. 

1,501 . . . log. 3.176381 
1,529 . . . log. 3.184407 

Cos. A, 10° 59' 9.991974 

GK=(R + r) sin. A. 

R + y, 1,529 . . . log. 3.184407 
Sin. A, 10° 59' . . . log. 9.279948 

GK = 291.3 2.464355 

GE = GK-^ COS. 1 A. 

GK, 291.3 . . . log. 2.464355 
Cos. i A, 5° 29i' . . . log. 9.998014 

GE = 292.6 2.466341 

GI = 2Rsm. iA. 

2 R, 1,910 . . . log. 3.281033 
Sin. -J A, 5° 29^ . . . log. 8.980916 

GI = 182.8 2.261949 

IE = GE — GI = 292.6 — 182.8 = 109.8. 

2. The distances GK and D, and one of the unequal 
radii, R, given, to find the other radius, r, and the central 
A. 



REVERSED CURVE WITH UNEQUAL ANGLES. Ufi 

Example. 
G K = 422, D = 30, R == 2,292. 

Taw. iA = D-^-GK. 

D = 30 . . . log. 1.477121 
GK = 422 . . . log. 2.625312 

Tan. i A, 40 04' . . . , . 8.851809 
.-.A = 8° 08'. 

GE = D-^sm. i A. 

D = 30 . . . log. 1.477121 
Sin. i A, 4° 04' . . . log. 8.850751 

GE = 423 2.626370 



GI = 2Rsm. iA. 



2 R = 4,584 . . . log. 3.661245 
Sin. i A, 4° 04' . . . log. 8.850751 

GI = 325.1 2.511996 

GE — G I = 423 — 325 = 98 = IE. 



r = i I E -^ sin. ^ A. 



iIE==49 . . . log. 1.690196 
-Sin. i A, 40 04' . . . log. 8.850751 

r = 691 2.839445 



XLI. 

A REVERSED CURVE HAVING UNEQUAL ANGLES. 

Given the angles A and B, and the length A B of a straight 
Jine connecting two diverging tangents, to find the radius of a 
leversed curve to close the angles. 

AI = R X tan. i A; B I = R X tan. \ B, 
.*. AB = R X {tan. ^ A + tan. \ B). 
,*. R = A B -h {tan. ^ A + tan. i BJ. 



120 REVERSED CURVE BETWEEN FIXED POINTS, 

Example. 
A = le**, B = 10°, A B = 840. 




AB, 840 log. 2.924279 

^ A = 8°, nat. tan. 0.14054 

i B = 5°, nat tan. 0.08749 

Tan. i A + tan. \B = 0.22803 . . . log. 1.357992 

K = 3684 3.566287 

This solution will apply also to the finding of the maximum 
radius for a simple curve which shall connect three tangents. 



XLII. 

A REVERSED CURVE BETWEEN FIXED POINTS. 

Given the angles N and K, and the length of the straight 
line E F connecting two divergent tangents, to find the radius 
of a reversed curve from E to F, connecting the tangents. 

1. Denote the angle i!l C or D I F by I; the angle CEI, 
complement of N, by n ; and the angle D F I, complement of 
K, by k. 

Then, in triangle E C I, — 

E C : C I : : sin. I : sin. n. . •. E C X sin. n = C I X sin. I. 



REVERSED CURVE BETWEEN FIXED POINTS. 



121 



Also, in triangle D F I, — 
DF : D I : : sin. I : sin. k. .'.DFX sin. fc = D I X s£w. L 

Adding these equations, we have — 

EC X sin. » + DF x sin. & = (CI + DI) X ««w. I. 




But E C and DF are each equal to K; sin. n = cos. N; sin, 
fc = COS. K; and C I + DI = 2 R. 
Hence the equation becomes, — 

R X {cos. N + COS. K) = 2 R X sin. I. 

.*. sin. I = {cos. N + COS. K) -f- 2. 

The foregoing elegant solution is abridged from Henck. 
2. Angle A = 180 — (?i + I) ; angle B = 180 — ( jk + I). 
To find radius, draw F H parallel, and E H perpendicular, to 
CD. 
Then E H = E F X sin. I. 

But EH = EG + GH; EG = RX sin. A; and GH = R 
X sin. B. 

.-. EF X sin. I = R X {sin. A + sin. B). 
.-. R = E F X sin. I -^ {sin. A -}- sin. B). 



122 REVERSED CURVE BETWEEN FIXED FOINTS, 

Example, 
E F = 1,400, N = 30°, K = 20°. 

Sin, I = {cos. N + COS. K) -^ 2. 

N = 30°, nat, cos 0, 

K = 20°, nat. cos 0. 



1.80572 
1.80572 -^ 2 = 0.90286 = nat. sin. 64° 32^ 
.-.I = 64° 32'. 

A = 180° — {n-\-l)= 180° — (60° + 64° 32') = 55° 28'. 
B = 180° — (A; + I) = 180° — (70° + 64° 32') = 45° 28'. 
K = E F X sin. I -H {sin, A + sin. B). 

EF = 1,400 log. 3.146128 

-ZVai. sm. I, 0.90286. . ]og. 1.955621 

EG = 1,264 .3.101749 

A = 55° 28' nat. sin 0.82380 

B = 45° 28' na«. sm 0.71284 

Sin. A + sin. B 1.53664 log. 0.186579 

R = 822.6 2.915170 

3. The young student should bear in mind that the addition 
or subtraction of the logarithms of two natural numbers gives 
a logarithm representing, not the sum or difference, but the 
product or quotient, of such numbers. When, therefore, as in 
the two foregoing cases, the sum or difference of two or more 
trigonometric functions — sines, tangents, and the like — is 
sought, the logarithm of the sum of the natural functions, and 
not the sum of their logarithms, is to be used. If, for example, 
sin. A X sin. B is required, the log. sin. A + log. sin. B = the 
logarithm of the product of the sines designated ; but, if sin. A 
-j- sin. B is sought, the natural sines of those angles must be 
added together, and the logarithm of the sum of these natural 
functions must be used in making logarithmic calculations. 



1 



TO CONNECl TWO DIVERGENT TANGENTS. 



123 



XLIII. 

TO CONNECT TWO DIVERGENT TANGENTS BY A 
REVERSED CURVE. 




1. ADVANCING TOWARDS THE INTERSECTION OF TANGENTS. 

Given the angle of divergence, N, tbc i.iitial P. C. at G, 
the distance GH, and the radii R, r, to find the central angles 
A and B. 

GK = C G X C08. N = R cob. N. 
GL = GH X sm. N. 

GK — GL = LKorEF, CF being drawn parallel to L E. 
Cos. B = D F -^ D C = (r + E F) -^ (R + r). 
Angle GC K = 90° — N ; angle D C F = 90° — B'. 
Angle A = GCK — DCF= (90° — N) — (90° — B) = B 
— N. • 

lExample. 
N = 24° 30', G H = 854, R = 1,440, r = 1,146. 

GK = R COS. N. 



R = 1,440 . 
Co8. N, 24° 30' . 



log. 3.158362 
log. 9.959023 



GK = 1,310 . . . . „ 3.117385 



124 



TO CONNECT TWO DIVERGENT TANGENTS. 



GL = GH X .sm. N. 

GH = 854 
Sin. N, 24° 30^ 



log. 2.931458 
log. 9.617727 



GL = 354 2.549185 

LKorEr = GK — GL = 1,310 — 354 = 956. 
Cos. B = (r + EF) -^ (R + r). 



r + EF = 2,102 . 

K + r = 2,586 . 

Cos. B, 35° 38' . 



log. 3.322633 
log. 3.412629 

. . 9.910004 



B = 35° 38'. 

A = B — N = 35° 38^ — 24° 30^ = 11° 08'. 




2. EECEDING FEOM THE INTERSECTION OF TANGENTS. 

Given the angle of divergence, N, the initial P. C. at G, 
the distance G H, and the radii R, r, to find the central angles 
A and B. 

GK = GH X «an. N. 
KC = GC — GK=-R — GK. 

LC orEF = KC X cos. N, the line C F being drawn paral- 
lel to L E. 
Cos. B = D F H- C D = (r 4- E F) -^ (R + r). 
Angle A manifestly = B -)- N. 



TO SHIFT A P. R. C. 125 

lElxam-ple. 
N = 18° 30', G H = 920, R = 955, r = 819. 

G K = G H X tan. N. 

GH = 920 . . . log. 2.963788 
Tan. 18° 30' . . . los. 9.524520 



GK = 307.8 2.488308 

KG = R — GK = 955 — 307.8 = 647.2. 
LCorEF = KC X cos. N. 

K C = 647.2 . . . log. 2.811039 

Cos. N, 18° 30' . . . log. 9.976957 

EF = 613.8 2.787996 



Cos. B = (r + EF) ^ (R + r). 

r + EF = 1,432.8 . . . log. 3.156185 

R 4- r = 1,774 . . . log. 3.248954 

Cos. B, 36° 08' 9^907231 



B = 36° 08 

A = B -f N = 36° 08' + 18° 30' = 54° 38^ 



XLIV. 

TO SHIFT A P. R. C. SO THAT THE TERMINAL 
TANGENT SHALL MERGE IN A GIVEN TANGENT 
PARALLEL THERETO. 

Given the reversed curve E F G, ending lu tangent GV: to 
find the angle of retreat, A, on the first branch, and the angle 
C of the second branch, ending in tangent PI T, parallel to 
GY. 

Measure the error T G = D, perpendicular to the terminal 
tangent. 



126 



TO SHIFT A P. R. C. 



In the figure, draw L K parallel to G Y, and passing through 
centre of first branch. 




Then M K = (R + r) X cos. B. 

NL ={R + >') X COS. C. 

WL = GK. 

KL =r + D-f GK. 

MK = r + GK. 

N L ~ M K = D. 

.-. (R + r) X COS. C — (R + r) X cos. B = D. 

.-. (R + r) X COS. C = (R + r) X cos. B + D. 

.-. Cos. C = [(K + r) X cos. B + D] -^ (R + r), 

A = (90° - C) - (90° - B) = B - C. 

Example. 
K = 1,433, r = 819, B = 34° 20^ D = 94. 
Cos. C = [(R + r) cos. B + D] -^ (R + r). 



K 4- r = 2,252 
B = 34° 20', cos. 



log. 3.352568 
log. 9.916859 



(R + r) cos. B = 1,860 3.269427 

Add D 94 



1,954 

(R + r) 



log. 3.290925 
log. 3.352568 



Cos. C, 29° 49/ 9.938357 

A = B — C = 340 20' — 29° 49' = 4° 31^ 



CURVE THROUGH A FIXED POINT. 



127 



XLV. 

I TO PASS A CURVE THROUGH A FIXED POINT, 
THE ANGLE OF INTERSECTION BEING GIVEN. 




Given the Intersection angle, A, of two tangents, to find the 
radius, R, of a curve which shall pass through a point, C; 
the position of said point, with reference to the tangents or the 
point of intersection, being known. 

1. By what data soever point C is located, they may be com- 
muted by simple processes to the form shown in the figure; 
namely, the ordinate BC and the distance IC to apex. Call 
the angle B I C a, and complete the triangle ICO. 



In this triangle, x = i 
Also, C O : I O : : sin. X 
ButCO = R: IO = 



180 



-\_a = 90°- (i 



A + a). 



sm. z. 
R 



COS. ^ A* 



.-.R: 



R 



COS. iA" 



sin. X : sm. z. 



Hence sin. z = 



COS. i A 
solved, and the radius ascertained 



The triangle ICO may then be 



128 



CURVE THROUGH A FIXED POINT. 



Exam2')le. 

A = 40°, B C = 32 feet, I B =- SO feet. 
Then BC -f- IB = 32 -=- 80 = 0.4 nat. tan. 21° 49'; 
I C = B C -^ nat. sin. 21° 49' = 32 ^ .372 = 86 feet. 
Also, a: = 90° — (i A + a) = 90° — 41° 49' = 48" 11'. 



and 



Next, sin. x, 48° 11' 
Divided by cos. i A, 20° 

= sin. z, 127° 31' 



log. 9.872321 
los. 9.97^ 



log. 9.899335 



Or, since the sine of any angle is equal to the sine of its sup- 
])lement, the supplement in this case, 52° 29', may be taken 
directly from the logarithmic table, from which supplement 
deducting x, or 48° 11', the remainder is the angle y = 4° 18'. 

Finally, IC = 86 . . . log. 1.934498 
Multiplied by sin. x, 48° 11' . . . log. 9.872321 

= CD . . . log. 1.806819 
And C D divided by sin. y, 4° 18' . . . log. 8.874938 

= C O = R = say, 855 feet . . . log. 2.931881 



2. In the case of a rectangular intersection, the solution is 
more simple. It is quite plain, from the figure, that — 

from which equation, 

R = a + 6 4- ^2ab. 




FROGS AND SWITCHES. 



129 



Example. 

a = 40, 6 = 80. 

Then R = 40 + 80 + VmOO = 200. 

3. Cases of this kind are disposed of with great ease in the 
field by means of the curve-protractor. 



XLVI. 



FROGS AND SWITCHES. 



t T W 


o 


l\ \>5>r AN 

/I .\r^^ 


/ 




V 





TO FIND THE RADIUS OF A TURNOUT CURVE, THE FROG 
ANGLES, AND THE DISTANCES FROM THE TOE OF SWITCH 
TO THE FROG POINTS. 

1. Draw the figure as in margin, C being the centre of the 
turnout curve, C K parallel to main track, and O K, I E, L M, 
perpendicular to it. Call the angle of the frogs at O, F; that 
of the intermediate frog at I, 2 F'; the throw of the switch-rail 
for single turnout, D; its angle with main track, S; the gauge 
of the track, G; and radius of outer rail, R. 

2. Usually the lengtli . and throw of switch-rail and the 
angles of the frogs at O are given. In that case, to find R, F' 
and U>o distances LO, LI, reason thu?:-' 



130 FROGS AND SWITCHES. 

3. The angle H N W, between the line of the switcli-rail pro- 
longed and a tangent to turnout curve at frog point O, = 
NOP — NIIW = F — S. The angle NOL or NLO, be- 
tween chord and tangent, = half the intersection angle H N W 
= i(F — S). The angle NOB = NOL + LOB. But NO L 
is seen to be = i (F — S), and NOB = F; then L OB = 
NOB — NOL = F — ^ (F — S) =i (F + S). The distance 
LO, from toe of switch to point of main frog, = LB -^ sin, 
LOB = (G — D) -^ sin. i (F + S). 

4. Again : the angle LCY = NLO=|(F — S);LY = i 
LO = i (G — D) -^ sin. i (F + S). LY -^ sin. LCY = 
LC; i.e., [^ (G — D) -f- sin. i (F + S)] -f- sin. i (F — S) 
= R. 

5. R may be found otherwise, as follows: — 

OK = OC co.s. KOC = Rcos. F; LM = LC co.s. CLM = 
Rcos. S; LM — OK = LB; i.e., R {cos. S — co.s. F) = (G — 
D). Hence R = (G — D) -^ {nat. cos. S — nat. cos. F). 

6. If R be known, to find F. This equation gives nat. cos. F 
= nat. cos. S — I (G — D) H- R]. 

1. To find the angle, 2 F', of the middle frog at L 

IE = I P + P E or O K; i.e., R cos. F' = i G + R co.s. F. 
Hence nat. cos. F' = nat. cos. F -\- (| G -4- R). 

8. The angle LI Y, by similar rciisoning to that used in rela- 
tion to LOB, is found to be = | (F' + S). The distance L I, 
from toe of switch to point of middle frog, = L Y -i- sin. L I Y 
= (iG — D)-^sm. i(F' + S). 

The preceding formulas translate into the following — 

RULES FOn FROG.S AND SWITCHES. 

9. To find the Angle of Switch-Rail with Main Track. 
Divide its throw, in decimals, by its length: the quotient 
will be the natural sine of the angle sought. 

10. To find the Distance from Toe of Switch to Point of 
Main Frog. 

Subtract the throw of switch-rail from the gauge of track, 
both in decimals; call the remainder a. Add together the 
angle of switch-rail with main track and the angle of the 
main frog; find the natural sine of half this sum, and call 
it 6. Divide a by b: the quotient will be the distance 
sought. 



FROGS AND SWITCHES. 131 

11. To find the Radius of Outer Bail of Turnout Curve. 
Subtract the throw of switch-rail from the gauge of track, 
both ill decimals ; call the remainder a. Subtract the natural 
cosine of the main frog angle from the natural cosine of the 
switch-rail angle; call the remainder h. Divide a by h: the 
quotient will be radius. 

12. To find the Main Frog Angle, the Badius of the Outer 

Bail being known. 
Call the natural cosine of the switch-rail angle a. Subtract 
the throw of switch-rail from the gauge of track, both in deci- 
mals. Divide the remainder by radius; call the quotient h. 
Subtract b from a: the remainder will be the natural cosine of 
the main frog angle. 

13. To find the Angle of the Middle Frog, in the Case of 

a Double Turnout. 
Call the natural cosine of the main frog angle a. Divide 
half the gauge of track by the radius of outer rail of turnout 
curve; call the quotient b. Add a and b together. Their sum 
is the natural cosine of half the middle frog angle. 

14. To find the Distance from Toe of Switch to Point of 

Middle Frog. 
Subtract the throw of switch-rail from half the gauge of 
track, both in decimals ; call the remainder a. Add together 
the switch-rail angle and half the middle frog angle. Find the 
natural sine of half this sum; call said natural sine b. Divide 
a by 6 : the quotient will be the distance sought. 

15. The use of logarithms will be found convenient in work- 
ing these rules. 

Examples. 

16. Switch-rail, 18 feet; throw, 5 inches = 0.42 feet; frog 
angle, 5° 44'; gauge, 4.71 feet. 

Sin. S = 0.42 -M8 = .02334 = sin. 1° 20'. 

LO = (G — D) -^ sin. |(F -f S) = (4.71 — 0.42) -f- sin. 
3° 32' = 4.29 -^ 0.0616 = 69.64 feet. 

K = (G — D) -H {nat. cos. S — nat. cos. F) = 4.29 -^ 0.00473 
= 907 feet. 

Nat. cos. F' = nai. cos. F -|- (| G -^ K) = 0.995 -f (2.354 -H 



232 FROGS AND SWITCHES. 

907) = 0.99759 = cos. 3° 58^. Hence the angle of Ihe middle 
frog = 2 F' = 7° 57'. 

LI = (i G — D) 4- sin. I [W + S) = (2.354 — 0.42) -^ sin. 
i (3° 58f + 1° 20') = 1.934 -h 0.0463 = 41.8 feet. 

17. In ordinary practice, frogs may be located with sufficient 
exactness by the following rules, deduced from the congruity 
of triangles. Great nicety in their location is not necessary. 
The important thing in practice is to lay the turnout curve so 
that the approach to the frog shall be fair and regular. How 
trackmen may do this without the use of instruments, in a 
very simple way, will be shown hereafter. Not that frogs may 
be set hap-hazard, and the approaches forced to fit: they 
ought to be nearly where they mathematically belong, and they 
can be thus placed by means of the rules subjoined. 

18. Let N stand for the number of the frog; 

L the length of switch-rail in feet; 
F the distance from toe of outer switch-rail to point 
of frog in feet. 
Then, for standard gauge, 4 feet 8^ inches, straight switch- 
rail, and 5 inches throw of switch. 

y_ 8.6LN 
L 4- 0.42 N* 

The above may be written roundly as a rule thus: — 
Multiply the length of switch-rail in feet by the number of 
the frog, and set down the product. Multiply that product by 
8^, and call the result A. Next add together the length of 
switch-rail in feet and two-fifths of the frog number; call the 
sum B. Then divide A by B, and the quotient will be the dis- 
tance in feet from toe of outer switch-rail to point of frog. 

Example. 
Switch-rail, 20 feet long; frog. No. 9. 

Length of switch-rail 20 

Multiplied by frog number .... 9 

Product 180 

Multiplied by . . 8^ 

1,530 = A. 

Length of switch-rail 20 

Added to | frog No, 9 3.6 



FROGS AND SWITCHES. 133 

A ilivided by B = 1,530 divided by 23.6 = 04,8 feet, the frog 
distance; say, (35 feet. 

19. If tlie switch-rail be curved, the formula would stand 
^lus: — 

8.6 LN 



F = 



L + 0.84 N 



Which may be made a written rule as follows: — 
Multiply the length of switch-rail in feet by the number of 
the frog, and their product by 8j; call the result A. Add 
together the length of switch-rail in feet and four-fifths of the 
trog number; call the sum B. Then divide A by B, and 
the quotient will be the distance from toe of outer switch-rail 
to point of frog in feet. 

20. The foregoing rules are applicable to turnouts from 
curves, as well as from straight lines. 

21. To find the radius of outer rail of a turnout curve from 
straight track. Data same as in previous rules for frogs; R 
'the required radius iu feet. 

8.6 L2 N2 



If the switch-rail be straight, R 
If the switch-rail be curved, R 



L2 — O.nN-^ 

8.6 L'^ N^ 
L2 — 0.68 W' 



22. To find the radius of the outer rail of a turnout curve 
from curved track, proceed thus : — 

First find the radius as for a turnout from straight track by 
I he preceding rule; call it, as before, R. Call the radius of the 
:nain track R2, and the required radius of turnout curve r. 

Then, if the turnout be towards the concave side of main 
track, — 

R. X R 
^ ~ R. + R* 

If the turnout be towards the convex side of main track, — 

Ro X R 

r = * . 

R.2 — R 

More explicitly, in the first case, r is equal to the product of 
the other radii divided by their sum ; and. In the second case, 
r is equal to the product of the other radii divided by thei: 
vlifference. 



134 FROGS AND SWITCHES. 

23. The angle of a frog is equal to 3,440' divided by the frog 
number. 

24. To find the frog distances and radii for a three-foot 
gauge, find them by the preceding rules for standard gauge, 
and take five-eighths of the result, using a switch-rail reduced 
\n like measure. 

For a metre gauge, take seven- tenths of the result, using a 
switch-rail reduced in like measure. 

Or these radii and distances may be found from the appended 
tables for standard gauge by pro-rating as above. 

25. Three frog patterns are enough for general service. 
They should be so proportioned, that, taken in couples, the less 
may fit as middle frogs on double turnouts. Numbers 5^, 7^, 
and lOi make an excellent suit; numbers 5, 7, and 9^- also 
answer very well. 

26. At the terminal stations, and about the shops of busy 
roads, patterns necessarily multiply. The better way in such 
cases is to plot the situation to a large scale, and to take the 
required distances and angles from the drawing. 



TURNOUT TABLE. 



135 







'Si 

11 


^1 
II 


s a" 
©to 




Main frog dist. 
Rad. outer rail. 
Mid. frog dist. 
Mid. frog angle. 


Main frog dist. 
■Rad. outer rail. 
Mid. frog dist. 
Mid. frog angle. 


Main frog dist. 
Rad. outer rail. 
Mid. frog dist. 
Mid. frog angle. 


Main frog dist. 
Rad. outer rail. 
Mid. frog dist. 
Mid. frog angle. 


Main frog dist. 
Rad. outer rail. 
Mid. frog dist. 
Mid. frog angle. 




32.6 

138.3 

21.3 

19° 16' 


.32.2 
138.5 

20.9 
19° 17' 


31.8 
138.8 

20.5 
19° 19' 


31.2 
139.4 

19.9 
19° 22' 


30.2 
140.6 

19.0 
19° 28' 


I- 


» 

H 
W 

> 

i 


36.4 
175.2 

23.7 
17° 08' 


35.9 

175.6 

23.2 

17"" 09' 


35.4 
176.0 

22.7 
17° 10' 


34.6 
176.9 

22.0 
17° 14' 


33.5 
178.9 

20.9 
17° 20' 


1- 


40.1 
215.6 

26.1 
15° 25' 


39.5 

216.6 

25.5 

15° 28' 


38.9 
217.4 

24.8 
15° 30' 


38.0 
218.8 

24.0 
1.5° 34' 


36.6 
221.8 

22.6 
15° 41' 


r 


^to||^ 

Sbbto 


43.5 
267.1 

27.8 
14° 00' 


42.7 
268.2 

27.1 
14° 02' 


41.6 
270.3 

26.0 
14° 06' 


39.9 
274.9 

24.4 
14° 14' 




■^OJCO-JI 


46.8 
314.2 

29.8 
12° 56' 


45.9 
315.7 

28.9 
12° 58' 


44.6 
318.6 

27.8 
13° 02' 


42.7 
325.0 

25.8 
13° 12' 


i» 


50.9 
362.0 

32.8 
11° 58' 


49.9 
363.7 

31.7 
12° 00' 


48.9 
365.8 

30.8 
12° 02' 


47.5 
369.7 

29.4 
12° 06' 


45.3 

378.4 

27.3 

12° 16' 


00 


54.9 
427.3 

34.2 
11° 04' 


05 50 05-^ 


52.6 
432.6 

32.8 
11° 10' 


50.9 
438.0 

31.3 
11° 14' 


48.4 
450.2 

28.9 
11° 24' 




58.1 
485.3 

37.0 
10° 25' 


56.8 
488.4 

35.6 
10° 28' 


55.5 
492.1 

34.5 
10° 30' 


53.7 
499.2 

32.8 
10° 34' 


50.9 
515.0 

30.1 
10° 46' 




61.8 
556.2 

39.3 
9° 42' 


60.3 
560.2 

37.9 
9° 44' 


58.8 
565.0 

36.5 
9°47' 


56.8 
575.0 

34.5 
9° 54' 


53.7 
595.8 

31.6 
10° 04' 


t-lOO 


65.0 
624.8 

41.3 
9° 10' 


63.4 
630.0 

39.7 
9° 14' 


61.8 
636.0 

38.1 
9° 16' 


59.5 
648.0 

35.9 
9° 22' 


66.1 
675.0 

32.8 
9° 34' 




68.7 
707.0 

43.3 
8° 39' 


66 8 
713.5 

41.6 
8° 41' 


65.0 
721.0 

39.9 
8° 44' 


62.6 
737.0 

37.4 
8° 51' 


58.8 
772.0 

34.0 
9° 03' 


=8 


72.0 
788.5 

15.2 
8° 14' 


70.0 
796.5 

43.3 
8° 17' 


68.0 
807.0 

41.4 
8° 21' 


65.3 

826.0 

38.7 

8° 27' 


61.3 

870.0 

35.1 

8° 40' 


i^ 


75.3 
875.0 

47.3 
7° 47' 


73.1 

885.0 

45.2 

7° 50' 


71.0 
897.0 

4.3.1 
7° 54' 


68.0 
921.0 

40.4 
8° 01' 


63.6 
976.7 

36.2 
8° 15' 


cn 


78.5 

964.2 

49.0 

7° 27' 


76.1 
976.2 

46.8 
7° 30' 


73.8 
992.0 

44.4 
7° 34' 


70.0 
1,020 

41.6 
7° 40' 


65.9 
1,089 

37.2 
7° 55' 


cn 

4S 


82.0 
1,068 

50.9 
7° 09' 


76.4 
1,083 

48.3 
7° 12' 


76.9 
1,102 

45.9 
7° 15' 


73.4 
1,137 
43.0 

7° 22' 


68.3 
1,224 
38.3 

7° 38' 


5^ 


84.8 
1,157 

52.9 
6° 50' 


82.0 
1,175 

49.9 
6' 54' 


79.4 
1,196 

47.6 
6° 57' 


75.7 
1,263 

43.8 
7° 09' 


70.3 
1,342 
39.2 

7° 21' 


cn 


J lm< 


■^*>.oooo 


82.5 
1.325 

49.0 
6° 41' 


78.6 
1,377 

45,3 
6° 49' 


72.7 
1,506 

40.1 
7° 05' 





136 



TURNOUT TABLE. 



o 

!z: 
<; 



"A 

m 
P3 




56.3 
4,117 

26.5 
8° 45' 


2P? 


looo tov 


Spl 


EpI 


si 


54.7 
3,135 

26.2 
8° 58' 


61.5 
1,762 

31.6 
8° 03' 


66.7 
1,469 

35.8 
7° 37' 


70.3 
1,343 

39,1 
7° 21' 


74.6 
1,256 

43.0 
7° 08' 

■ 




53.4 
2,516 

25.9 
9° 10' 


59.9 
1,549 
31.1 

8° 18' 


64.6 
1,315 

35.1 
7° 51' 


68.2 
1,215 

38.3 
7° 36' 


72.2 
1,144 

41.9 
7° 24' 


SI 


25.5 
9° 26' 


58.1 
1,351 

30.4 
8° 36' 


62.6 
1,169 

34.3 
8° 10' 


65.8 
1,090 
37.3 

7° 55' 


69.6 
1,032 

40.7 
7° 43' 


s5: 


50.6 
1,672 

25.1 
9° 42' 


56.4 
1,182 

29.8 
8° 54' 


60.5 
1,040 

33.5 
8° 29' 


63.6 
976.8 

36.3 
8° 15' 


67.1 
930.3 

39.5 
8° 03' 


"°: 


49.0 
1,370 

24.6 
10° 04' 


54.4 
1,022 

29.1 
9° 16' 


59.1 
914.7 

32.5 
8° 53' 


61.2 
865.2 

35.1 
8° 39' 


64.4 

828.3 

38.1 

8° 28' 




47.5 
1,160 

24.1 
10° 26' 


52.6 
894.7 

28.4 
9° 38' 


56.2 
811.1 

31.6 
9° 16' 


58.8 
772.1 

34.1 
9° 03' 


61.8 
742.8 

36.8 
8° 53' 


.:4 


45.8 
947.3 

23.6 
10° 52' 


50.4 

767.0 

27.6 

10° 08' 


53.7 
704.9 

30.5 
9° 46' 


56.2 
675.2 

32.8 
9° 34' 


58.8 
652.7 

35.3 
9° 24' 


>5 
00 "-1 


44.1 
798.2 

23.0 
11° 18' 


48.5 

666.2 

26.8 

10° 36' 


51.5 

618.9 

29.5 

10° 16' 


53.7 
595.8 

31.6 
10° 04' 


56.2 
578.1 

33.9 
9° 55' 


"-t 


42.2 
656.1 

22.3 
11° 54' 


46.1 

564.3 

25.8 

iri4' 


48.8 
529.8 

28.2 
10° 56' 


50.9 
513.0 

30.2 
10° 44' 


53.0 
499.7 

32.2 
10° 36' 


t-r-l 
°00 


40.4 
5.53.9 

21.7 
12° 30' 


44.0 
487.1 

24.9 
11° 52' 


»OC0(M^ 


48.4 
448.3 

28.9 
11° 23' 


50.3 
438.3 

30.8 
11° 14' 


CO 


38.3 
451.1 

20.9 
13° 18' 


41,5 
405.7 

23.8 
12° 42' 


43.7 
387.6 

25.8 
12° 26' 


45.3 
378.4 

27.3 
12° 16' 


47.1 
371.2 

29.0 
12° 08' 


CO^ 

°o 


36.4 
375.8 

20.1 
14° 08' 


39.3 
343.7 

22.7 
13° 34' 


41.3 
330.7 

24.6 
13° 18' 


42.7 
324,0 

25.9 
13° 10' 


44.2 
318,6 

27.4 
13° 02' 




34.4 
310.2 

19.3 
15° 06' 


36,9 
288,1 

21.6 
14° 36' 


38.7 
278.9 

23.2 
14° 21' 


39,9 
274.0 

24.4 
14° 12' 


41.2 
270.3 

25.7 
14° 06' 


< 


31.9 
244.9 

18.2 
16° 30' 


34.1 

230.8 

20.2 

16° 02' 


35.6 
224.9 

21.6 
15° 48' 


36.6 
221.8 

22.6 
15° 41' 


37.7 
219.3 

23.7 
15° 35' 


4 


29,5 
193,7 

17.1 
18° 06' 


31.4 

184.8 

18.8 

17° 40' 


32.6 
181.0 

20.0 
17° 28' 


33,5 

178,9 

20.8 

17° 20' 


34.4 

177.3 

21,7 

17° 15' 


■i 


26.9 

149.6 

15.8 

20° 08' 


28.5 
144.2 
17.29 
19° 44' 


29.5 
141.9 
18.25 
19° 34/ 


30.2 
149.6 

18.9 
19° 28' 


31.0 

139.6 

19.7 

19° 23' 




Main frog dist. 
Rad. outer rail. 
Mid. frog dist. 
Mid. frog angle. 


Main frog dist. 
Rad. outer rail. 
Mid. frog dist. 
Mid. frog angle. 


Main frog dist. 
Rad. outer rail. 
Mid. frog dist. 
Mid. frog angle. 


Main frog dist, 
Rad. outer rail. 
Mid. frog dist. 
Mid. frog angle. 


Main frog dist. 
Rad. outer rail. 
Mid, frog dist. 
Mid, frog angle. 




Switch-rail, 
L'gth. 12 ft. 
Ang. 4° 00'. 
Rad. 171.9. 


Switch-rail, 
L'eth. 16 ft. 
Ang. 3° 00'. 
Rad, 312.7. 


Switch-rail, 
L'gth. 20 ft. 
Ang. 2° 24'. 
Rad. 477.5. 


Switch-rail, 
L'gth. 24 ft. 
Ang. 2° 00'. 
Rad. 687.6, 


Switch-rail, 
L'gth, 30 ft, 
Ang, 1° 36', 
Rad. 1,074. 



TO LOCATE A TURNOUT, 



137 



XLvn. 

TO LOCATE A TUENOUT. 

\. Let the heavy parallels in the figure represent the rails of 
the main track. 




2. Stick a pin or drive a spike at A, the toe of switch, aV, 
a distance from the gauge side of the main-track rail equal to 
the throw of the switch-rail. Lay off the distances A C and 
AB (if a double turnout), taken from the foregoing tables, and 
place the frogs C and B, or mark those points. Stretch the 
cord from A to B, and from B to C. Mark the middle points 
of those stretches at H and P. Catch the cord at H with your 
forefinger, and pull it outwards until your finger, at E, lines 
with the switch-rail, and also with the right gauge side of frog 
B, Stick a pin at L, half-way between H and E. Let the 
cord spring in against L, so that it shall stretch straight from 
A to L, and from L to B. Opposite the middle points, V, of 
those stretches, stick pins on the outside at a distance from 
the cord equal to one-quarter of H L. In like manner, catch 
the cord at P, the point midway between B and C ; stretch it 
to F, in line with the gauge sides of the fro^s ; and §tick a pip 
^X I, half -way between J* aud Ft 



138 TO LOCATE A TURNOUT. 

3. Next lay off the proposed line of the near rail of the sid$ 
track, X D. Mark the point G on that line where the range 
of the proper gauge side of frog C strikes it. Measure C G. 
Set off G D, equal to C G, along the side-track line, and drive 
a pin at D. Stretch the cord from C to D. Mark the middle 
point of it at K, and drive a pin at N, half-way between K and 
G. Stretch the cord from C to N, and from N to D. Stick 
pins outside the middle points, M and O, of those stretches at 
a distance from those points, M and O, equal to one-quarter of 
KN. 

4. These three sets of pins will fix the line of one rail of the 
turnout. The corresponding rail of a double turnout can be 
laid off from them, if required, by symmetrical measurements. 

5. In the case of a single turnout, stretch the cord from the 
toe of switch, as above, to the point of frog, located by the 
foregoing tables ; catch it at the middle, and pull it outwards 
to a point in range with the line of the switch-rail in one 
direction, and the gauge side of frog in the other direction. 
Half-way between that point and the middle of the cord, when 
straight, stick a pin. Measure that half-way distance, and 
divide it by 4; call the quotient the "quarter-distance." 
Stretch the cord from the pin just set to the toe of switch in 
one direction, and to the point of frog in the other. Outside 
the middle points of these short stretches, lay off the " quarter- 
distance," as above found, and stick two other pins. These 
three pins will sufficiently mark the line of the outer rail of 
Ihe turnout. 

6. The same methods will apply in practice to turnouts from 
curves. In the latter case, the distance C G is to be calculated 
as follows : — 

Multiply the distance Y D, between the nearest rails of the 
parallel tracks, by the number of the frog, taken from the fore- 
going table. Thus, on the full gauge, with a space between 
tracks of 7 feet and a No. 6 frog, the distance C G would be 7 
X 6 = 42 feet. Lay off C G, in range of the gauge side of the 
frog, and stick a pin at G. Measure out GD, equal to CG, 
and set another pin at D, making D Y the proper distance be 
tween tracks. Then stretch the cord from D to C, and pro 
ceed to stake off the curve C N D, as above directed. 



CROSSINGS ON STRAIGHT LINES, 



139 



XLVIII. 

CROSSINGS ON STRAIGHT LINES. 

1, Having located frogs B and C by the preceding methods, 
stretch the cord any convenient distance, C D, in the range of 




the outer gauge side of the frog C. Set off E F parallel to 
CD, and distant the gauge-width from it. The intersection of 
said parallel at F with the near rail of the side track marks 
the spot for point of side-track frog; the curve F G, thence to 
toe of switch, corresponds to A C on the main track, and may 
be staked out in like manner. 



XLIX. 



CROSSINGS ON CURVES. 

1. Having located frogs B and C by the preceding methods, 
set off the width of gauge, C D, from point of frog C, and 
square to its outer gauge side. Stick a pin at D. 

2. Next calculate the distance D E to the point of side-track 
frog as follows: Subtract the gauge of track from the dis- 



/40 



CROSSINGS ON CURVES. 



tance, H I, between the gauge sides of the nearest rails of the 
main and side tracks ; multiply the remainder by the number 




of the frog, taken from foregoing tables. The product will be 
the distance from D to the point of side-trftck frog at £. 



ELEVATION OF THE OUTER RAIL ON CURVES. 14J 

3. Suppose, for example, the gauge sides of the nearest rails 
of the main and side tracks are 6 feet 6 inches asunder; gauge 
of track, 4 feet Si inches; frog, a No. 9. Kediici ug inches to 
dechnals, we have then the distance between tracks 6,5 feet, 
less the gauge, 4.7 feet, = 1.8 feet; and 1.8 multiplied by i), 
the number of the frog, gives 16.2 feet for the distance D E. 
The proper spring will be given to rail DE on the ground; and 
curve E G, from frog to toe of side-track switch, will be staked 
off as directed in the section on turnouts. 



L. 
ELEVATION OF THE OUTER RAIL ON CURVES. 

1. Great precision in this adjustment is unattainable, owiiij^- 
to differences in the speed of trains and to the cost of track - 
maintenance, if it were attempted. 

2. Molesworth gives the following formula for determining 
the elevation of the outer rail with any gauge : — 

V — greatest velocity of trains in feet per second. 
G — gauge of railway in feet. 

C = length of chord whose middle ordinate will give the 
required elevation. 

Then C = i V ^ G^ 

A modification of 
this formula gives the 
following approximate 
rules : — 

To fix the elevation 
of the outer rail on the standard gauge of 4 feet ^\ inches, 
multiply the speed of trains in miles per hour by 5, and divide 
the product by 3. This will give the length of tape, C, to 
stretch on the gauge side of the outer rail; and the distance, c. 
from the middle of the tape to the gauge side of the rail, wi.l 
be the proper elevation. 

For guage of one metre, — 3.28 feet, make C equal to one 
and one-third times the speed of trains in miles per hour. 

For 3-feet gauge, make C equal to one and one-fourth time 
the speed of trains in miles per hour. 




143 ELEVATION OF THE OUTER RAIL ON CURVES. 



TABLE OF ELEVATIONS OP OUTER RAIL ON CURVES. 

This table was formulated by the writer from Pennsylvania Rail Road 
practice as follows : 

(jV speed in miles per hour + 1) X (by the degree of curve) = elevation of 
outer rail expressed in 8ths of an inch. 





SPEED IN MILES PER HOUR. 


10 


20 


30 


40 


50 


60 








fi 




VALUES IN EIGHTHS OF AN INCH. 




2° 


4 


6 


8 


10 


12 


14 


4° 


8 


12 


16 


20 


24 


28 


6° 


12 


18 


24 


30 


36 


42 


8° 


16 


24 


32 


40 


48 




10° 


20 


30 


40 


50 






12° 


24 


36 


48 








14° 


28 


42 










16° 


32 


48 




•• 


•• 





Note.— The limit of elevation of outer rail is 6^ inches. 



TRACKMEN'S TABLE OF CURVES. 



14< 



LI. 



ocooo-ac»tgi*^cot0h-^otooo~jcgi:^ifjcotoi-j 


DEGREE OP CURVE. 




OOMO.OI-'rfs.ollibOWQOI-itOCntOOCOaiCD^ 
ts4^ iUt9 COtOtOK) 


DEFLECTION 

DISTANCES 

IN FEET 

AND INCHES. 


2 1-2 
5 1-4 
8 
10 1-2 
13 

15 3-4 
18 1-4 
21 

231-2 
261-4 
29 

311-2 
34 

36 3-4 
39 1-4 
42 

441-2 
47 1-4 
49 3-4 
52 1-2 


1 


3 

■ I 

4 

'4 

2 « 

H W 

|. 

S 

IS 

o 

H 

n 


'■.■''r'T"r''r'T"r"r'7"r"7-'T"r"r" "r"?^ 'r"T" 

oo to 00 io to 00 ifi. to 00 *>. to 00 *> 00 tt-oo it-oo 




*.*»*^*^(MI»COMtatOtOtOI--MI-.i-i 

tut £S£g|£SSS£2|||S^ 


8 


• 

COtOtOtOtOtOlCi-'Mp-'l-'f-'l-'H- 

t-'OiOi H-"^ OSOJ 050J 




1-8 

3-16 

5-16 

3-8 

1-2 

5-8 

3-4 

13-16 

15-16 

11-16 

11-8 

11-4 

11-2 

15-8 

13-4 

17-8 

1 15-16 

2 1-16 

2 5-16 


1 


Ft. In. 
463 5 
328 5 
266 2 
232 
207 7 
189 8 
175 6 
164 1 
154 10 
146 6 
140 
134 1 
129 
124 1 
120 
116 2 
112 10 
109 7 
106 8 
104 


►S£q 

ff 


LENGTH OF CHORD IN FEET AND 

INCHES, WITH A MIDDLE 
ORDINATE EQUAL TO GAUGE OP 
TRACK. 


|.JtSOJp-'l-'©tOl-'t>OOt>0-»©tO*«.|-»M-^0© P 


f 


^^^^^^!^^^^^«S°g-J®o«^.'^«o'i^ 


DEGREE OP CTJRVB. 



X44 trackme:n's table of curves: 



EXPLANATION OF THE FOREGOING TABLE. 

Columns 1 and 10 give the degree of curve. 

The use of column 2, containing the deflection distances, 
may be illustrated thus : Suppose stakes 4, 5, and 6 to be miss- 
ing from a 3-degree curve, and that stakes 2 and 3 are still 
standing 100 feet apart. To replace the missing stakes, pro- 
ceed as follows: Measure 100 feet from 3 to A, and make a 
mark at A exactly in range with 2 and 3. Find, in column 2 
of the table, the deflection distance for a 3-degree curve, which 
is seen to be 5 feet 3 inches. Hold one end of the tape at A, 




and, stretching 5 feet 3 inches towards 4, nearly square to the 
range A-3, make a scratch on the ground three or four feet 
long, swinging the tape around A as a centre. Next lay off 
100 feet from stake 3 to the scratch; where the end of that 
measurement strikes it, is the place for stake 4. By measuring 
100 feet out to B on the range 3-4, and proceeding in like 
manner, stake 5 may be set ; and so on. 

3. If the centre line is already staked for track at points 100 
feet asunder, and the degree of curve is wanted, range out the 
straight line between stakes, as above, to A or B, and measure 
across from those marks to the neighboring location-stake. 
Suppose the distance B-5, for example, to be 8 feet 9 inches. 
Referring, then, to column 2 of the table, we find that deflec- 
tion distance to indicate a 5-degree curve. If the distance 



TRACKMEN'S TABLE OF CURVES. 



145 



proved to be 4 feet 4 inches, we should soon discover that that 
distance was about half-way between 3 feet 6 inches and 5 feet 
3 inches, the nearest numbers in the table corresponding 
respectively to a 2-degree and a 3-degree curve, and showing 
the located line to be a 2^-degree curve. 

-1. Let A C B in the figure, which is drawn very much out of 
proportion in order to make the subject clear, represent the 
centre line of a curve. Suppose G H to be a chord 100 feet 
long, and G C or C H to be a chord 50 feet long. Then column 
3 in the table gives the distance, C D, from the middle of the 
100-feet chord to the rail, and column *4 gives the distance, 
E F, from the middle of the 50-feet chord to the rail, for the 
different degrees of curve. By the aid of these columns, pins 
can be set 25 feet apart on a curve where the location-stakes 
are 100 feet apart. Thus, for a 3-degree curve, C D is 8 inches, 



N ^ N 




and E F 2 inches. If pins were wanted at the half-way marks^ 
'^, their distance from the dotted short cliords would be one>= 
quarter of E F. It must be an uncommon case, however, that 
calls for stakes closer togetlier than 25 feet. 

5. Columns 5, 6, and 7 give the spring of rails of different 
lengths for the various degrees of curve. 

6. Columns 8 and 9 give figures for finding the degree of 
curve, by simple measurement of a straight line on the track, 
as follows : Suppose A C B and K I L to represent the rails of a 
curving track. From any point A, on the outer rail, sight 
icross to a point B, on the same rail, along a line just touching 
the inner rail at I. Measure from A to B, and seek the dis- 
tance in column 8 or 9, according to the gauge of track. If 
the distance, for example, measured 232 feet on the full gauge, 
then the curve would be a 4-degree curve ; if 249 feet, then it 
would indicate a 3i-degree curve, for the reason that the 



146 TRACKMEir*S TABLE OF CURVES. 

measured distance falls half-way between the distances corre- 
sponding to a 3-degree and a 4-degree curve respectively. 

7. The rate of curve can be found also very nearly by means 
of column 3. To do so, stretch a straight line, 100 feet long, 
between points on either rail ; for, though they seem very dif- 
ferent in the figure, the two rails of a track have practically 
the same curvature. Measure from the middle of the line 
across to the gauge side of the rail, and seek the measured 
distance in column 3: opposite to it, in column 1, will be 
found the degree of curve. 

8. If, in any case, the exact figures sought are not found in 
the table, take out the next figure less and the next greater. 
Subtract one from the other, and divide the remainder by 4. 
Add the fourth part of the difference between them, thus 
determined, to the smaller number, and compare the sum with 
the number sought. If still too small, add another fourth 
part; and so on until the distance or the degree is ascertained 
to within a quarter part. 

9. Suppose, for instance, a deflection distance measures 5 
feet 7 inches. The nearest tabular numbers are 5 feet 3 inches 
and 7 feet. Their difference is 21 inches, one-fourth of which 
is 5j inches. Adding h\ inches to the smaller number, 5 feet 
3 inches, gives 5 feet 85- inches, which indicates nearly enough 
a 3|^-degree curve. Again: if a measurement of 175 feet is 
sought in column 9, the track is see::> at once, witbo»at calcmat- 
tion, to be a 4^-degree curve. 



TABLES 



TABLES OF THE TIMES OF CULMINATION AND 

OF ELONGATION OF TEE POLESTAll AND 

OF ITS AZIMUTH AT ELONGATION. 

These tables are designed to facilitate the determination of a 
meridian line and of the magnetic declination (variation of 
the compass) by simple instrumental means (p. 44). For this 
purpose the tables are sufficiently accurate. They will also 
be found useful when preparing for or laying out work for 
a more refined determination of the astronomical azimuth 
and for the measures of the value of an eye-piece micrometer. 



148 



TABLE I. 



MEAN LOCAL (ASTRONOMICAL) TIM^., COUNTED FROM NOON 
AND FROM ZERO TO TWENTY-FOUR HOURS, OF THE 
CULMINATIONS AND ELONGATIONS OF POLARIS IN THE 
YEAR 1889. COMPUTED FOR LATITUDE 40° NORTH AND 
LONGITUDE 6 HOURS WEST FROxM GREENWICH. 



1889. 


Date. 


E. Elong. 


LTppER Culm. 


W . Elong. 


Lower Culm. 




h. m. 


h. m. 


h. m. 


h. m. 


Jan. 1 

" 15 


36.2 


6 31.0 
5 35.7 


12 25.7 
11 30.4 


18 29.1 
17 33.8 


23 37.0 


Feb. ] 


22 29.9 


4 28.6 


10 23.3 


16 26.7 


15 


21 34.6 


3 33.3 


9 28.1 


15 31.4 


March 1 


20 39.4 


2 38.1 


8 32.8 


14 36.2 


15 


19 44.4 


1 43.1 


7 37 7 


13 41.1 


April 1 
" 15 


18 37.4 
17 42.4 


36.0 


6 30.7 
5 35.7 


12 34.1 
11 39.0 


23 37.1 


May 1 


16 39.5 


22 34.2 


4 32.9 


10 36.1 


" 15 


15 44.6 


21 39.3 


3 38.0 


9 41.2 


June 1 


14 37.9 


20 32.7 


2 31.3 


8 34.6 


15 


13 43.0 


19 37.8 


1 36.4 


7 39.7 


July 1 
15 


12 40.4 
11 45.5 


18 35.2 
17 40.3 


33.8 


6 37.1 
5 42.2 


23 35.0 


Aug. 1 


10 39.0 


16 33 8 


22 28.4 


4 35.7 


" 15 


9 44.1 


15 38.9 


21 33.5 


3 40.8 


Sept. 1 


8 37.5 


14 32.3 


20 26.9 


2 34.2 


" 15 


7 42.6 


13 37.4 


19 32.0 


1 39.3 


Oct. 1 
15 


6 39.7 

5 44.7 


12 34.5 
11 39.5 


18 29.1 
17 34.1 


36.4 


23 37.6 


Nov. 1 


4 87.9 


10 32.7 


16 27.3 


22 30.8 


15 


3 42.7 


9 37.5 


15 32.2 


21 35.6 


Dec. 1 


2 39.7 


8 34.5 


14 29.2 


20 32.6 


15 


1 44.4 


7 39.2 


13 34.0 


19 37.3 



To refer the tabular times to any year subsequent to the 
tabular year (1889) add 0"\33 for every year. 

To refer the tabular times, corrected as above, to any year 
in a quadrennium, observe the following rules: 

For the first year after a leap-year the table is correct. 

For the second year after a leap-year add 0™.9 to the tabular 
value. 

For the third year after a leap-year add 1™.7 to the tabular 
value. 

For leap-year and befoi'e March 1 add 2'". 6 to the tabular 
value. 

For leap-year from and after March 1 subtract 1™.2 from the 
tabular value. 



CULMINATIOXS AND ELONGATIONS OF POLARIS. 149 



To refer to any calendar day other than the 1st and 15th of 
each mouth, subtract 3'". 94 for every day between it and the 
preceding tabular day, or add 3'". 94 for every day betweeu it 
and the succeeding tabular day. 

The longitude correction will amount to 0"'.16for each hour. 

To refer to any other than the tabular latitude, and between 
the limits of 25° and 50° North, add to the time of west elouga 
tion 0°i.l3 for every degree South of 40° and subtract from the 
time of west elongation 0™.18 for every degree North of 40°. 
Reverse these signs for corrections to times of east elongation. 

Observe that the year 1900 is not a leap-year, and this must 
be kept in view when dealing with dates from and after 
March 1 of that year. The 20th century begins after the ex- 
piration of Dec. 31, 1900. 

The deduced tabular times may be relied on to have no 
greater error than ± 0*^.3. 

Table II. below Lat. 24° is abridged from a table for each 
degree of latitude between 25° and 50° North, computed for 
this book by Mr. C. A. Schott, Asst. Supt. of the U. S. C. and 
G. Survey, with the mean declination of Polaris for each 
year. A closer result will be had by applying to the tabular 
values the following correction, which depends on the differ- 
ence of the mean and the apparent places of the star : 



For 
Middle of 


Lat. 25° 


Lat. 40° 


Lat. 50° 


For 
Middle of 


Lat. 25° 


Lat. 40° 


Lat. 50° 


Jan. 

Feb. 

March 

April 

May 

June 


- 0'.3 
-0.3 

- 0.1 
0.0 

+ 0.2 
+ 0.3 


- 0'.4 
-0.3 

- 0.2 
0.0 

+ 0.2 
+ 0.3 


-0'.4 
- 0.4 
-0.2 
0.0 
+ 0.2 
+ 0.3 


July 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 


+ 0'.2 

+ 0.1 

0.0 

- 0.2 

- 0.5 
-0.6 


+ 0'.3 
+ 0.1 
-0.1 
-0.3 
-0.6 
-0.8 


+ 0'.3 
+ 0.2 

- 0.1 

- .3 
-0.7 

- 0.9 



The deduced tabular azimuth, counted from the North, 
may generally be depended upon with no greater error than 
± 0'.2. 

In making the computation the mean places of Polaris were 
first accurately deduced from Newcomb's Catalogue of 1098 
standard clock and zodiacal stars, Washington, 1881, for five 
equidistant epochs. From these fundamental places those f(jr 
each year were readily found by interpolation. 

Azimuth for latitudes less than 25° was reckoned by the 
author from the data for that degree. 



150 



AZIMUTH OF POLARIS AT ELONGATION. 





• 03 




Xi *" 




a XL 




^5 




.2 H 




^ 


H 






.3 5 


cc 


d S 


H 


^ a 


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2 >» 2 



g 


n 
o 










s 




t4 


s 


S 


eg 


'& 

-^ 


1 


03 


H 


^ 


53 


>^ 


OJ 






,C3 


QJ 


H 




-d 


s 


t; 


*^ 


H 


fl 


d 




rt 




O 


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S 


t^ 


p 


s 


§ 


d 


-1 




;_i 


H 




CS 


3 








3 
bO 

d 

c3 


W 


+-' 


A 








H 


<u 


o3 


^ 


P. 


03 ^ ■ 


^ 


^ 


3;;^ 


M 
W 


OS 


-^f 


fS 


03 


^ CO 


02 


QJ 


03 -^ 


S 


d 


03 iO 


2 


■T3 

d 

bo 


Xi O 

>i 

y-i 


O 


d 


w 


1 


1? 


^ 


o 


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s 


n-t 


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03 


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1 =dS2SS28?!S?SS§SSS^'^g 


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30rrS0X>C0»O00C000i0i0CDO«ii0<.-0»(NJ0 


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SS^'^$^Sg55S^^=;jg^53S^^5^S 


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1 






1 


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S;2i2^S5J§I^S^gJS^^g^^SS 


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sss2s^§ijs^^8S?j§^^i;^s?s 


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i 


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SS^SS?:j^^^7^5;S?i§^^^5SL5 


1 


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SS$^S§?5;5^^S55S^^^iS§S?S 


i 




1 


osTj<Tj'oo<»ojeoQO»o-<s'«oQoeoi-ic*«o«n?oco 


J 


o ^«'ss§s;^^8^;s^^§^^^^g 



TABLES. 



n\ 



TABLE III. 



ROODS AND PERCHES IN" DECIMAL PARTS OF AN ACRE. 
One Acre=4 Roods = 160 Perchcs = 4,840 Square Yards =43,560 Square Feet. 



w 
I 

OS 

u 




Roods. 




u 
s 
u 

w 

Oh 
21 




Roods. 







1 


3 


3 


« i 


1 


3 


3 


o 


.000 


.250 


.500 


750 


.131 


381 


.631 


881 


I 


.006 


.256 


.506 


756 


22 


•137 


3«7 


.637 


887 


2 


.012 


.262 


.512 


752 


23 


.144 


394 


.644 


894 


3 


.019 


.269 


•519 


769 


24 


.150 


400 


.650 


900 


4 


.025 


.275 


•525 


775 


25 


.156 


406 


.656 


906 


5 


.031 


.281 


•531 


781 


26 


.162 


412 


.662 


912 


6 


.037 


.287 


•537 


787 


^l 


.169 


419 


.669 


919 


7 


.044 


.294 


•544 


794 


28 


•175 


425 


•675 


925 


8 


.050 


.300 


•550 


800 


29 


.181 


431 


.681 


931 


9 


.056 


.306 


•556 


806 


30 


.187 


437 


.687 


937 


lO 


.062 


.312 


.562 


812 


3^ 


.194 


444 


.694 


944 


II 


.o6q 


• 319 


•569 


819 


32 


.200 


450 


.700 


950 


12 


.07s 


.325 


■575 


82s 


33 


.206 


456 


.706 


956 


13 


.081 


•331 


.581 


B31 


34 


.212 


462 


.712 


Q62 


14 


.087 


•337 


.587 


^Z7 


35 


.219 


469 


.719 


969 


15 


.094 


•344 


•594 


844 


36 


.225 


475 


•725 


975 


16 


.100 


•350 


.600 


8^0 


37 


.231 


481 


•731 


981 


17 


.106 


•356 


.606 


856 


38 


•237 


487 


•737 


987 


t8 


.112 


.362 


.612 


862 


39 


•244 


494 


•744 


994 


19 


.119 


•369 


.619 


869 


40 


■ 250 


Soo 


•750 I 


000 


20 


.135 


•375 


.625 


«75 















TABLE IV. 

DECIMALS OF AN ACRE IN ONE CHAIN LENGTH OF 100 FEET, 
AND OF VARIOUS WIDTHS. 









Width in 
Rods. 


Dec 

Ac 








i Width in 
1 Rods. 


Decimals of an 

Acre per 100 

Feet. 


Acres per 
Mile. 


imals of an 
re per 100 
Feet. 


Acres per 
Mile. 




i V2 


.018939 


I 


5^ 


■208333 


II 




1 


.037879 


2 


6 




227273 


12 




iH 


.056818 


3 


6^ 




246212 


13 




2 


•075757 


4 


7 




265151 


14 




. 2j^ 


.094697 




1% 




284091 






3 


.113636 


6 


8 




303030 


16 




3^ 


.132576 


7 


8M 




321970 


17 




4 


•151515 


8 


9 




340909 


18 




4^ 


.170454 


9 


9^ 




359848 


19 




5 


.189394 


10 


10 


.378788 


20 









152 



TABLES. 



TABLE V. 

ACRES, ROODS, AND PERCHES IN SQUARE FEET. 



Acres. 


Square Feet. 


Roods. 


Square Feet. 


Perches. 


Square Feet. 


I 


43560 


I 


10890 


17 


4628.25 


2 


87120 


2 


21780 


18 


4900.50 


3 


130680 


3 


32670 


19 


5172-75 


4 


174240 


4 


43560 


20 


5445.00 


5 
6 

I 


217800 
261360 
304920 
348480 






21 
22 
23 


5717-75 
5989-50 
6261.75 
6534.00 


Perches. 


Square Feet. 






24 


9 


392040 


I 


272.25 


25 


6806.25 


lO 


435600 


2 


544-50 


26 


7078.50 


II 


479160 


3 


816.75 


27 


7350.75 


12 


522720 


4 


1089.00 


28 


7623.00 


13 


566280 




1361.25 


29 


7895-25 


14 


609840 


6 


1633.50 


30 


8167.50 




653400 


7 


1905-75 


31 


8439-75 


16 


696960 


8 


2178.00 


32 


8712.00 


17 


740520 


9 


2450.25 


33 


8984-25 


18 


784080 


10 


2722.50 


34 


9256.50 


19 


827640 


11 


2994-75 


35 


9528.75 


20 


871200 


12 


3267.00 


36 


9801.00 


21 


914760 


13 


3539-25 


37 


10073.25 


22 


958320 


14 


3811.50 


38 


10345-50 






15 


4083-75 


39 


10617.75 






16 


4356.00 


40 


10890.00 





TABLE VL 

CIRCULAR ARCS TO RADIUS OF 1. 



■ - — i 


Degrees. 


Minutes. 


Seconds. 


I 
2 
3 
4 


.01745329 
.03490658 
-05235988 
.06981317 


/ 

2 
3 
4 


.00029089 
.00058178 
.00087266 
.00116355 


// 

2 
3 
4 


.00000485 
.00000970 
.00001454 
.00001939 


5 


.08726646 


5 


.00145444 


5 


.00002424 


6 


.10471975 


6 


.00174533 


6 


.00002909 


7 
8 

9 


.12217305 
.13962634 
.15707963 


I 

9 


.00203622 
.00232711 
.00261799 


7 
8 

9 


^l-l! 





TABLES. 



153 



TABLE VII. 

FEET m DECIMALS OF A MILE. 





Feet. 


Decimals of a Mile. 


I 
2 
3 
4 
5 
6 

7 
8 

9 


O.O 00189394 

0.0 00378798 

0.0 00568182 
o.\) 00757576 
0.0 00946970 
0.0 01 136364 
0.0 01325758 
0.0 I 5 I 5 I 5 2 
0.0 01704546 



TABLE YIIL 

INCHES REDUCED TO DECIMAL PARTS OF A FOOT. 



In. 





I 


2 


3 


4 


5 


6 


, 


C 


9 


10 


II 


In. 


!? 


.0000 


■0833 


.1667 


.2500 


•3333 


.4167 


.5000 


•5833 


.6667 


.7500 


•8333 


.9167 


! 

1 


, 1^ 


.0052 


•0855 


.1719 


•2552 


.3385 


.4219 


•5052 


.5885 


.6719 


•7552 


.8385 


.9219 


T^n 


' i 


.0104 


.0938 


.1771 


.2604 


•3438 


.4271 


.5104 


•5938 


.6771 


.7604 


.8438 


.9271 


i 


i'^ 


.0156 


.0990 


.1823 


.2656 


•3490 


•4323 


•5156 


.5990 


.6823 


.7656 


.8490 


•9323 


-'-^ \ 


* 


.0208 


.1042 


.1875 


.2708 


■3542 


■4375 


.5208 


.6042 


.6875 


.7708 


-8542 


•9375 


i 1 


jiA 


.0260 


.1094 


.1927 


.2760 


•3594 


■4427 


.5260 


.6094 


.6927 


.7760 


■8594 


•9427 


V'b 1 


' * 


.0313 


.1146 


.1979 


.2813 


.3646 


■4479 


•5313 


.6146 


.6979 


•7813 


.8646 


•9479 


' 


1 -h 


.0365 


.1198 


.2031 


.2865 


.3698 


•4531 


•5365 


.6198 


.7031 


.7865 


.8698 


•9531 


1'6 i 


' i 


.0417 


.1250 


.2083 


.2917 


•3750 


•4583 


•5417 


.6250 


-7083 


•79^7, 


.8750 


•9583 


^ 


1 A 


.0469 


.1302 


•2135 


.2969 


.3802 


•4635 


.5469 


.6302 


•7135 


.7969 


.8802 


•9635 


U I 


* 


.0521 


•1354 


.2188 


.3021 


•3854 


.4688 


■5521 


.6354 


.7188 


.8021I 


.8854 


.9688 


5 1 


•}i 


•0573 


.1406 


.2240 


•3073 


.3906 


.4740 


•5573 


.6406 


.7240 


.8073 


.8906 


.9740 


H 


i 


.0625 


.1458 


.2292 


•3125 


•3958 


.4792 


•5625 


.6458 


.7292 


.8125I 


.8958 


•9792 


i i 


\l 


0677 


.1510 


.2344 


.3177 


.4010 


■4844 


•5677 


.6510 


•7344 


•8177 


.9010 


9844 


II i 


I 


•or29 


•1563 


.2396 


•3229 


.4063 


.4896 


•5729 


.6563 


•7396 


.8229 


•90631 


.9896 


' ; 


H 


.0781 


1615 


.2448 


.3281 


.4115 


•4948 


•5781 


.6615 


•7448 


.8281 


9"5 


9948 


il 



TABLE IX. 

RADII AND THEIR LOGARITHMS, MIDDLE ORDI- 

NATES, AND DEFLECTION DISTANCES. 

Note. — This table applies to chords of lOOfeet units. To 
use it for chords of 20-metre units divide the degree of the 
proposed metric curve by 2 and tiud the quotient in the first 
column of the table; or multiply any tabular degree of curve 
by 2 for the degree of the proposed metric curve. Then one- 
tenth of the radius opposite the tabular degree thus found or 
assumed will give very nearly the true radius in metres of the 
proposed metric curve, and four-tenths of the corresponding 
figures in the last three columns will give very nearly the 
values of those functions in metric measure. Thus, if a 3° 
metric curve be proposed, we take one-tenth of the radius 
opposite the tabular degree 1° 30', or 382 metres, and find the 
values of the functions in the last three columns to be .1308, 
1.048, and ,524. Resulting errors may be neglected except in 
work of extraordinary accuracy. 



156 



RADII AND THEIR LOGARITHMS. 



Degree 




Logarithm 


Arithmetical M 


iddle 


Deflec- 


Tangen- 1 
tialDis- 1 


of 


Radius. 


of 


Comple- Ore 


inate, 


tion Dis- 


Curve. 




Raduis. 


merit. C 
100 


hord 
Feet. 


tance. 


tance. 1 


o / 












1 


5 


68754.9 


4.837304 


5.162696 


.018 


•145 


•073 ! 


lO 


•34377-5 


4.536274 


5.463726 


.036 


.291 


•145 1 


IS 


22918.3 


4.360182 


5.639818 


•055 


-436 


.218 ! 


20 


17188.8 


4.235246 


5.764754 


•073 


.582 


.291 


25 


13751-0 


4-138335 


5.861665 


.091 


.727 


•364 


30 


11459.2 


4-059154 


5-940846 


.109 


•873 


• 436 


35 


9822.2 


3.992209 


6.007791 


.127 


1.02 


-509 


40 


8594.4 


3-934215 


6.065785 


•145 


1.16 


-582 / 


45 


7639-5 


3.883066 


6.116934 


.164 


1-31 


-654 


1 50 


6875-5 


3-837304 


6.162696 


.182 


1-45 


.727 


1 SS 


6250.5 


3-795914 


6.204086 


.200 


1.60 


.800 


1 


5729.6 


3.758128 


6.241872 


.218 


1^75 


.873 


i 5 


5288.9 


3-723365 


6.276635 


.236 


1.89 


•945 


1 10 


4911.1 


3.691179 


6.308821 


•255 


2.04 


1.02 


i IS 


4583-7 


3.661216 


6.338784 


•273 


2.18 


1.09 


j 20 


4297-3 


3.633195 


6.366805 


.291 


2.33 


1. 16 


25 


4044.5 




6.393136 


•309 


2.47 


1.24 


30 


3819-8 


3.582041 


6.417959 


•327 


2.62 


1-31 


1 35 


3618.8 


3-558565 


6-441435 


•345 


2.76 


1.38 


1 40 


3437-9 


3.536293 


6.463707 


.364 


2.91 


1.45 


45 


3274-2 


3-515106 


6.484894 


.382 


3-05 


1.53 


50 


3125.4 


3-494906 


6.505094 


.400 


3.20 


1.60 


' 55 


2989.5 


3-475599 


6.524401 


.418 


3.34 


1.67 


; 3 


2864.9 


3.457114 


6.542886 


-436 


3-49 


1.74 


s 


2750-3 


3.439380 


6.560620 


-455 


3-64 


1.82 


10 


2644.6 


3-422359 


6.577641 


•473 


3-78 


1.89 


'"> 


2546.6 


3.405961 


6-594039 


.491 


3-93 


1.96 


! 20 


2455 -7 


3-390175 


6.609825 


-509 


4.07 


2.04 


li 25 


2371.0 


3-374932 


6.625068 


.527 


4.22 


2.ir i 


! 30 


2292.0 


3.360215 


6.639785 


.545 


4-36 


2.18 1 


35 


2218. I 


3-345982 


6.654018 


•564 


4-51 


2.25 1 


40 


2148.8 


3.332196 


. 6.667804 


.582 


4-65 


2.33 


45 


2083.7 


3-318835 


6.681165 


.600 


4.80 


2.40 


1 so 


2022.4 


3-305867 


6.694133 


618 


4.94 


2.47 


1 55 


1964.6 


3-293274 


6.706726 


636 


5-09 


2.54 


3 


1910.1 


3.281056 


6.718944 


655 


5-23 


2.62 1 


i 5 


l8q8.5 


3.269163 


6.730837 


673 


5.38 


2.69 


10 


1809.6 


3.257584 


6.742416 


691 


5-53 


2.76 


! IS 


1763-2 


3.246301 


6.753699 


709 


5-67 


2.84 


20 


1719.1 


3-235301 


6.764699 


727 


5-82 


2.91 


1 25 


1677.2 


3-224585 


6.775415 


745 


5.96 


2.98 


30 


1637-3 


3.214129 


6.785871 


764 


6.11 


3.05 


35 


1599.2 


3.203902 


6.796098 


782 


6.25 


3-13 


40 


1562.9 


3.193931 


6.806069 


800 


6.40 


3-20 


45 


1528.2 


3.184180 


6.815819 


818 


6.54 


3-27 


50 


1495.0 


3. 1 74641 


6.825359 


836 


6.69 


3-34 


55 


1463.2 


3-165303 


6.834607 


855 


6.83 


3-4^ 


4 


1432.7 


3-156155 


6.843845 


873 


6.98 


3-49 


5 


1403.5 


3.147212 


6.852788 


891 


7.12 


3-56 


10 


1375-4 


3.138429 


6.861571 


909 


7.27 


3-63 


15 


1348.4 


3.129819 


6.870181 


927 


7.42 


3-71 


20 


1322.5 


3-121395 


6.878605 


945 


7-56 


3-78 


25 


1297.6 


3.113141 


6.886859 


964 


7.71 


3-85 


30 


1273.6 


3-105033 


6.894967 


982 


7-85 


3-93 


35 


1250.4 


3.097048 


6.902952 I, 


00 


8.00 


4.00 


40 


1228.1 


3-089233 


6.910767 I. 


02 


8.14 


4.07 





RADII JJfD THEIR LOGARITHMS. 



157 



Degree 




Logarithm 


Arithmetical 


Middle 


Deflec- 


Tangen- 


of 


Radius. 


of 


Comple- 


v_/rQmatej 
Chord 


tion Dis- 


tial Dis- 1 


Curve. 

o / 
4 45 




Radius. 


ment. 


100 Feet. 


tance. 


tance. j 


1206.6 


3.081563 


6.918437 


1.04 


8.29 


4.14 i 


50 


1185.8 


3.074011 


6.925989 


1.05 


8-43 




22 


55 


1165.7 


3.066587 


6.933413 


1.07 


8.58 




29 1 


5 


1146.3 


3.059299 


6.940701 


1.09 


8.72 




36 1 


5 


1127.5 


3.052117 


6.947883 


I. II 


8.87 




43 \ 


TO 


1109.3 


3.045050 


6.954950 


1-13 


9.01 




51 i 


15 


1091.7 


3.038103 


6.961897 


1-15 


9.T6 




58 I 


20 


1074.7 


3.031287 


6.968713 


1. 16 


9-30 




65 ! 


25 


1058.2 


3.024568 


6.975432 


1. 18 


9-45 




72 ; 


30 


1042.1 


3.017910 


6.982090 


1.20 


9.60 




80 1 


35 


1026.6 


5.011401 


6.988399 


1 .22 


9-74 




87 i 


40 


1011.5 


3.004967 


6.995033 


1.24 


9-89 




94 1 


45 


996.9 


2.998652 


7.001348 


1-25 


10. 




02 1 


50 


982.6 


2.992377 . 


7.007623 


1.27 


10.2 




09 ! 


55 


968.8 


2.986234 


7.013766 


1.29 


10.3 




16 


G 


955-4 


2.980185 


7.019815 


1. 31 


10.5 




23 


5 


942.3 


2.974189 


7.025811 


1-33 


10.6 




31 


10 


929.6 




7.031704 


1-35 


10.8 




38 


^5 , 


917.2 


2.962464 


7-037536 


1.36 


10.9 




45 


20 


905.1 


2.956697 


7-043303 


1-38 


11.0 




52 


25 


893.4 


2.951046 


7.048954 


1.40 


11.2 




60 ! 


30 


882.0 


2.945469 


7-054531 


1.42 


11-3 




67 : 


35 


870.8 


2.939918 


7.060082 


1-44 


II-5 




74 1 


40 


859.9 


2.934448 


7-065552 


1-45 


11.6 




81 1 


45 


849-3 


2.929061 


7.070939 


1-47 


11.8 


5 


89 \ 


50 


839.0 


2.923762 


7.076238 


1-49 


II. 9 


5 


96 ! 


55 


828.9 


2.918502 


7.082498 


1-51 


12. 1 


6 


03 1 


7 


819.0 


2.913284 


7.086716 


1-53 


12.2 


6 


10 ! 


5 


809.4 


2.908163 


7.091837 


1-55 


12.3 


6 


18 1 


10 


800.0 


2.903090 


7.096910 


1-56 


12.5 


6 


25 { 


15 


790.8 


2.898067 


7-101933 


1.58 


12.6 


6 


32 I 


20 


78T.8 


2.893096 


7.106904 


1.60 


12.8 


6 


39 ! 


25 


773-1 


2.888236 


7.111764 


1.62 


12.9 


6 


47 ! 


30 


764-5 


2.883377 


7. I 16623 


1.64 


131 


6 


54 1 


35 


756.1 


2.878579 


7.121421 


1.65 


13.2 


6 


61 1 


40 


747-9 


2.873844 


7.126156 


1.67 


13.4 


6 


68 1 


45 


739-9 


2.869173 


7.130827 


1.69 


■13-5 


6 


76 1 


50 


732.0 


2.864511 


7-135489 


1-71 


13-7 


6 


83 


55 


724-3 


2.859918 


7.140082 


1-73 


13.8 


6 


90 


8 


716.8 


2.855398 


7.144602 


1-75 


14.0 


6 


98 


5 


709.4 


2.850891 


7.149109 


1.76 


14.1 




05 ! 


10 


702.2 


2.846461 


7-153539 


1.78 


14.2 




12 1 


15 


695.1 


2.842047 


7-157953 


1.80 


14-4 




19 i 


20 


688.2 


2.837715 


7.162285 


1.82 


14-5 




27 


25 


681.3 


2-833338 


7.166662 


1.84 


14.7 




34 


30 


674.7 


2.829111 


7.170889 


1.85 


14.8 




41 1 


35 


668.1 


2.824841 


7-175159 


1.87 


15.0 




48 i 


40 


661.7 


2.820661 


7-179339 


1.89 


15-1 




56 1 


45 


655-4 


2.816506 


7.183494 


1.91 


15.3 




63 


50 


649-3 


2.812445 


7-187555 


1-93 


15-4 




70 


55 


643.2 


2.808346 


7.181654 


1-95 


15-5 




'' 1 


9 


637-3 


2.804344 


7-195656 


1.96 


15-7 




1 
85 


5 


631.4 


2.800305 


7.199695 


1.98 


15-8 




92 


10 


625.7 


2.796366 


7.203634 


2.00 


16.0 


7 


99 


15 


620.1 


2-. 792462 


7.207538 


2.02 


16.1 


8.06 II 


1 



158 



RADII AND THEIR LOGARITHMS. 





Degree 




Logarithm 


Arithmetical 


Middle 


Deflec- 


Tangen- 
tial Dis- 


of 


Radius. 


of 


Comple- 


Ordinate, 


tion Dis- 


Curve. 




Radius. 


ment. 


Chord 
100 Feet. 


tance. 


tance. 


9 20 


614.6 


2.788593 


7.211407 


2.04 


16.3 


■ 1 
8.14 


25 


60.^. I 


2.784689 


7.215311 


2.06 


16.4 


8.21 


j 30 


603.8 


2.780893 


7.219107 


2.07 


16.6 


8.28 


1 35 


598.6 


2.777137 


7.222863 


2.09 


16.7 


8.35 


1 40 


593-4 


2.773348 


7.226652 


2.11 


16.8 


8.43 


45 


588.4 


2.769673 


7.230327 


2.13 


17.0 


8.50 


50 


583.4 


2.765966 


7.234134 


2.15 


17. 1 


8.57 


55 


578.5 


2.762303 


7.237697 


2.16 


17-3 


8.64 


10 


573-7 


2.758685 


7.241315 


2.18 


17.4 


8.72 


10 


564.3 


2.751510 


7.248490 


2.22 


17.7 


8.86 


20 


555.2 


2.744449 


7.255551 


2.26 


18.0 


9.00 


30 


546.4 


2-7375" 


7.262489 


2.29 


18.3 


9.15 


40 


537-9 


2.730702 


7.269298 


2-33 


18.6 


9.30 


50 


529.7 


2.724030 


7.275970 


2.36 


18.9 


9.44 


11 


521.7 


2.717421 


7.282579 


2.40 


19.2 


9-58 


10 


513.9 


2.710879 


7.289121 


2.44 


19.5 


9-73 


20 


506.4 


2.704494 


7.295506 


2.47 


19.7 


9-87 


30 


499.1 


2.698188 


7.301812 


2.51 


20.0 


10. 


40 


492.0 


2.691965 


7.308035 


2.55 


20.3 


10.2 


50 


485.1 


2.685831 


7.314169 


2.58 


20.6 


10.3 


18 


478.3 


2.679700 


7.320300 


2.62 


20.9 


10.4 


10 


471.8 


2.673758 


7.326242 


2.66 


21.2 


10.6 


20 


465.5 


2.667920 


7.332080 


2.69 


^^'l 


10.7 


30 


459.3 


2 . 662096 
2.656386 


7-337904 


2.73 


21.8 


10.9 


40 


453.3 


7.343614 


2.77 


22.1 


II. 


50 


447.4 


2.650696 


7.349304 


2.80 


22.4 


11.2 


13 


441.7 


2.645127 


7.354873 


2.84 


22.6 


II-3 


10 


436.1 


2.639586 


7.360414 


2.88 


22.9 


11-5 


20 


430.7 


2.634175 


7-365825 


2.91 


23.2 


II. 6 


30 


425.4 


2.628797 


7.. 371 203 


2.95 


23.5 


11.7 


.40 


420.2 


2.623456 


7-376544 


2.98 


23.8 


II. 9 


50 


415.2 


2.618257 


7.381743 


3.02 


24.1 


12.0 


14 


410.3 


2.613102 


7.386898 


3.06 


24.4 


12.2 


10 


405.5 


2.607991 


7.392009 


3-09 


24.7 


12.3 


20 


400.8 


2.602928 


7.397072 


3.13 


25.0 


12.5 


30 


396.2 


2.597914 


7.402086 


3.t7 


25.2 


12.6 


40 


391.7 


2.592954 


7.407046 


3.20 


25.5 


12.8 


50 


387-3 


2.588047 


7.411953 


3-24 


25-8 


12.9 

1 


15 


- 383-1 


2.583312 


7.416688 


■ 3-28 


26.1 


13.0 


10 


378-9 


2-578525 


7.421475 


3-3^ 


26.4 


13.2 


20 


374-8 


2.573800 


7 . 426200 


3.35 


26.7 


13-3 


30 


370.8 


2.569140 


7.430860 


3-39 


27.0 


13.5 


40 


366.9 


2.564548 


7.435452 


3.42 


27.3 


13.6 


50 


363-0 


2-559907 


7.440093 


3.46 


27-5 


13.8 


16 


359.3 


2-555457 


7.444543 


3.50 


27.8 


13.9 


10 


355-6 


2.550962 


7.449038 


3.53 


28.1 


14. 1 


20 


352.0 


2.546543 


7-453457 


3-57 


28.4 


14.2 


30 


348.4 


2.542078 


7.457922 


3.61 


28.7 


14-3 


40 


345.0 


2.537819 


7.462181 


3-^^ 


29.0 


14.5 


50 


341.6 


2.533518 


7.466482 


3.68 


29.3 


14.6 


17 


338.3 


2.529302 


7.470698 


3.72 


29.6 


14.8 


10 


335.0 


2.52504s 


7-474955 


3.75 


29.9 


14.9 



RADTI AN-D THEIR LOGARITHMS. 



159 













Deflec- 


Tangen- 


Degree 




Logarithm 


Arithmetical 


Middle 


of 


Radius. 


of 


Comple- 


Ordinate, 


tion Dis- 


tial Dis- i 


Curve, 




Radius. 


ment. 


Chord 
100 Feet. 


tance. 


tance. 


o / 












j 


17 20 


33^-8 


2.520876 


7.479124 


3-79 


30.1 


15-1 


30 


328.7 


2.516800 


7.483200 


3-82 


30-4 


15-2 


40 


325-6 


2.512684 


7.487316 


3-86 


30.7 


^5-4 


50 


322.6 


2 . 508664 


7-491336 


3-90 


31.0 


15-5 


18 


319.6 


2.504607 


7-495393 


3-93 


3T.3 


T5-6 


10 


316.7 


2.500648 


7-499352 


3-97 


31.6 


15.8 


20 


313-9 


2.496791 


7.503209 


4.01 


31-9 


15-9 


30 


311-1 


2.492900 


7.507100 


4.04 


32.1 


16. 1 


40 


308.3 


2.488974 


7.5T1026 


4.08 


32-4 


16.2 


50 


305-6 


2-485153 


7.514847 


4.12 


32-7 


16.4 


19 


302.9 


2.481299 


7.518701 


4-15 


33-0 


T6.5 


10 


300.3 


2-477555 


7-522445 


4.19 


33-3 


16.6 


■20 


297.8 


2.473925 


7-526075 


4-23 


33-6 


16.8 


30 


295.2 


2.470116 


7.529884 


4.26 


33-9 


16.9 


40 


292.8 


2.466571 


7-533429 


4-30 


34-2 


17. 1 


50 


290.3 


2.462847 


7-537153 


4-34 


34-4 


17.2 


30 


287.9 


2.459242 


7-540758 


4-37 


34-7 


17.4 


1_ . __l 



TABLE X.— (See p. 160.) 
FOR USE WITH A 20-METRE CHAIN. 

Engineers accustomed to thinking their degree of curvature 
with reference to the 100-ft. chain may find it convenient to 
remember that the degree of curvature, if a 20-metre chain be 
used, is approximately tico-thirds of the foregoing. Thus a 
3° metric curve would be about equivalent to a ^r^'' curve laid 
out with the 100-ft. chain. 

A 20-metre chain = 65.618 feet ; a 100-ft. chain = 1.524 
chains of 20 metres each, one foot being equal to 0.3048 of a 
metre, and a metre equal to 3.2809 feet. 

If a metric curve is to be retraced with a 100-ft. chain, the 
exact degree of curvature should be ascertained with reference 
to the radius in feet, as set forth in Art. XVIII. 

It is convenient to mark stakes with the even numbers, 
2, 4, 6, etc., when using the 20-metre chain, distance being 
thus recorded in tens of metres. 



160 



METRIC CURVE TABLE. 



Degree 


Radius 


Loga- 


1 
Arithme- Mi 


J.Ord. 


Deflec- Ta 


ngen- 


of 


in 


rithm of 


ticalCom- C 


hord 


tion Dis- tia 


1 Dis- 


Curve. 


Metres. 


Radius. 


plement. 20 ]^ 


Metres. 


tance, tc 


ince. 


lO 


6875.50 


3-837304 


6.162696 


0076 


.0582 


0291 


20 


3437-75 


3-536274 


6 


463726 


0144 


.1164 


.0582 


30 


2291.84 


3-360184 


6 


639816 


0218 


.1745 


0873 


40 


1718.88 


3-235246 


6 


764754 


0290 


.2327 


1 1 64 


50 


I375-II 


3-138338 6 


861663 


0363 


.2909 


1454 


1 


1145-93 


3.059158 6 


940842 


0437 


-3491 


1745 


10 


982.23 


2.992213 7 


007787 


0509 


-4072 


2036 


20 


859-46 


2.934226 




065774 


0582 


■4654 


2327 


30 


763-97 


2.883076 




1 1 6924 


0655 


.5236 


2618 


40 


687.57 


2.837317 




162683 


0727 


.5818 


.2909 


50 


625,07 


2.795929 




204071 


0800 


-6399 


3200 


2 


572.99 


2.758147 




241853 


0873 


.6981 


3490 


10 


528.92 


2 • 723390 




276610 


0945 


-7563 


3781 


20 


491.14 


2.691205 




308795 


1018 


-8144 


4072 


30 


458.40 


2 661245 




338755 


1091 


.8726 


4363 


40 


429.76 


2.633226 




366774 


1:64 


.9308 


• 4654 


50 


404.48 


2.606897 




393103 


1237 


.9889 


4945 


3 


382.02 


2.582086 




417914 


1309 


1.047 


5235 


10 


361.91 


2.558601 




440399 


1382 


1 .105 


5526 


20 


343-82 


2-536331 




463669 


1454 


1. 163 


.5817 


30 


327-46 


2-515158 




484842 


1527 


1.222 


6108 


40 


312.58 


2.4H961 




505039 


1600 


1.280 


6398 


50 


298.99 


2 475657 




524343 


1673 


1.338 


6689 


4 


286.54 


2.^57185 




542815 


1746 


1.396 


6980 


lO 


275.08 


2-4.39459 




560541 


i8i8 


1-454 


7^71 


20 


204 51 


2.422442 




577558 


1891 


1.512 


7561 


30 


254-71 


2 . 406046 




59J954 


1964 


1-570 


7852 


40 


245.62 


2.390264 




609736 


2036 


1 . 629 


8143 


50 


237 16 


2.375041 




624959 


2109 


1.687 


bH 


5 


229.26 


2.360328 




639672 j 


2182 


1-745 


8726 


20 


214.94 


2.332317 




667683 1 


2328 


1. 861 


9308 


40 


202 . 30 


2.305996 




694004 


2473 


1-977 


9889 


6 


191.07 


2.280193 




719807 


2619 


2.093 I 


047 


20 


i8r.o3 


2-257751 




742249 


2764 


2.210 I 


105 


40 


171 98 


2 235478 




764522 


2910 


2.326 I 


163 


7 


163.80 


2.214314 




7S5686 


3055 


2.442 I 


222 


30 


156.37 


2.194153 




805847 


3201 


2-558 1 


280 


40 


149.58 


2.174874 




825,26 


3347 


2.674 I 


338 


8 


143-36 


2.156428 




843572 ! 


3492 


2.790 I 


396 


20 


137-63 


2.138713 




861287 


363S 


2 906 I 


454 


40 


132.35 


2 121724 




878276 


3783 


3 . 022 I 


512 


9 


127-45 


2. 1053 to 




894660 


3929 


3.138 I 


570 


20 


122. 9 t 


2.089587 




910413 


4075 


3-254 1 


^o^ 


40 


118.68 


2.074378 




925622 


4220 


3-370 I 


687 


10 


114.74 


2 059715 




940285 


4366 


3486 I 


745 


33 


109.29 


2.038580 




961420 


4585 


3.660 I 


832 


11 c 


104.33 


2.018409 




981591 


4803 


3834 


919 


30 


99-81 


I. 999174 


8 


000826 


5022 


4.008 2 


006 


12 


95-67 


1.980776 


8 


019224 


5241 


4.181 2 


093 


30 


91.86 


I. 963126 


8 


036S74 


5460 


4 355 2 


i8x 


13 


88.34 


I. 946157 


8 


053843 


5679 


4528 


268 


30 


85.08 


1.920828 


8 


070172 


5897 


4.701 2 


355 


14 


82.06 


1.914132 


8.085868 


6117 


4-875 2 


442 



TABLE XI. 

SQUARES, CUBES, ETC., OF NUMBERS 
FROM 1 TO 1042. 

Note. — If N be taken to represent any number in any 
column of this table, then the algebraic significance of the re- 
maining numbers, on the same line, in terms of N, will be as 
given in the following synopsis : 



N 


N2 


N^ 


i^N 


fN 


1 

N 


|/N 


N 


VN^ 


fN 


fN 


1 

4/N 

1 

fN 


Vn 


fN^ 


N 


'fN 


fN 


N2 


W 


N6 


N 


fN^ 


1 

N2 

1 

X3 


IS-3. 


K6 


N9 


i/W 


N 


1 

N 


1 

N=^ 


1 

N^ 


»4 


^1 


N 



TABLE 



SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS 



.. 


Squaws. 


Cubes. 


Square Roots. 


Cube Roots. Reciprocals. 


1 


1 


1 


10000000 


1-ooooono 


100000000 


2 


4 


8 


1-4142136 


12599210 


500000000 


3 


9 


27 


1-7320508 


1-4422496 


.333333333 


4 


16 


64 


2-0000000 


1-5874011 


250000000 


5 


25 


125 


2-2360680 


1-7099759 


200000000 


6 


36 


216 


2-44948;t7 


1-8 1712. (6 


166666667 


7 


49 


343 


2-6457513 


1-9129312 


142857143 


8 


64 


512 


2-8284271 


2-OOOOUOO 


125000000 


9 


81 


729 


3-0000000 


2-0800837 


111111111 


10 


100 


1000 


3-1622777 


2-1544347 


100000000 


il 


121 


1331 


3-3166248 


2-2239801 


090909091 


12 


144 


1728 


3-4641016 


2-2894288 


083333333 


13 


169 


2197 


3-6055513 


2-3513347 


076923077 


14 


196 


2744 


3-7416574 


2-4101422 


071428571 


15 


225 


3375 


3-8729833 


2-4602121 


066666667 


IG 


256 


4096 


4-0000000 


2-5198421 


062500000 


17 


289 


4913 


4-1231056 


2-5712816 


058823529 
05.55o5556 


18 


324 


5832 


4-2423407 


2-02j7414 


J9 


3f)l 


6859 


4-3588989 


2-6684016 


052631579 


20 


400 


8000 


4-4721360 


2-7144177 


050000000 


21 


411 


9261 


4-5825757 


2-7589-243 


047619048 


22 


484 


10648 


4'6904158 


2-8020393 


045454545 


23 


529 


12167 


4-7958315 


2-8438670 


(^3478261 


24 


570 


13824 


4-8989795 


2-8844991 


041666667 


25 


625 


15625 


5-0000000 


2-9240177 


040000000 


26 


676 


17576 


5-0990195 


29624960 


038461538 


27 


729 


19683 


5-196152.4 


3-0000000 


037037037 


28 


784 


21952 


5-2915026 


30365889 


035714236 


29 


841 


24389 


5-385 1G48 


3-07-23168 


034482759 


30 


O'M) 


27000' 


5-4772256 


3-1072325 


033333333 


31 


9;ii 


29791 


5-56:7044 


3-14138U6 


032258065 


32 


1024 


32708 


5-6568542 


3-1748021 


031250000 


33 


lum 


35937 


5-744>>l)26 


3-2075343 


0303(»;<030 


34 


1156 


39304 


5-8309.:49 


3-2396118 


029411765 


35 


1225 


42875 


5-91 1)0798 


3-2710663 


0285714-29 


3(i 


1296 


46656 


6-0000000 


3-3019272 


027777778 


37 


1369 


50653 


6-0827625 


3-3322218 


027027027 


38 


1444 


54872 


6-1644140 


3-3619754 


026315789 


39 


1521 


59319 


6-2449980 


3-3912114 


025641026 


40 


1000 


64000 


6-3245553 


3-4199519 


025000000 


41 


1681 


68921 


6-4031242 


3-4482172 


024390244 


42 


1764 


74088 


6-4807407 


3-4760266 


023809524 


43 


1849 


70507 


6-5574385 


3-5U33981 


023255814 


44 


1936 


85184 


6-6332496 


3-5303483 


022727273 


45 


2025 


91125 


6-7082039 


3-556a933 


02222-2-222 


46 


2116 


97336 


6-7823300 


3-5830479 


021739130 


47 


2209 


103823 


6H55t)546 


3-6088261 


021276600 


48 


2304 


110592 


6-9282032 


3-634-2411 


020833333 


49 


2401 


117649 


7-0000000 


3-6593057 


020408163 


$0 


2500 


mm 


7-0710678 


3-68403H 


02000WOO 



T63 



SQUARES, CUBES, ETC., OF NUMBERS. 



163 



No. 


Squarea, 


Cubes. 


Square Roots. 


Cube RooU. 


Reciprocali. 


51 


2601 


132651 


7-1414284 


3-7084298 


•019607843 


.•58 


2704 


140608 


7-2111026 


3-7325111 


-019230769 


53 


2809 


148877 


7-2801099 


3-7.562858 


•018867925 


54 


2916 


157464 


7-3484692 


3-7797631 


-018518519 


5a 


3025 


160375 


7-4] 6 1985 


3-8029525 


-018181818 


56 


3136 


175616 


7 4833148 


3-8258624 


•0178.57143 


57 


3249 


185193 


7-5498.344 


3-8485011 


•017.543860 


58 


3364 


195112 


7-6157731 


3-87087(i6 


•017241.379 


59 


3481 


205379 


7-6811457 


3-8929905 


•016949153 


61» 


3600 


216000 


7-7459667 


3-9148670 


•016666667 


61 


3721 


226981 


7-8102497 


3-93o4;i72 


-01639.3443 


6i 


3^44 


238328 


7-87401)79 


3-9578^15 


-016129032 


6:< 


3969 


250047 


7-9372539 


3-9790.571 


•01587.3016 


64 


4096 


262144 


8-0000000 


4-00000 JO 


-01562.5000 


65 


4225 


274625 


8-0622577 


4-02J7256 


-01.5384615 


66 


4356 


287496 


8-1240384 


404J-2401 


-015151515 


67 


44H9 


300763 


8-1853.128 


4-0615480 


•014925373 


68 


4624 


314432 


8-2462113 


4-0816551 


-01470.5882 


69 


4761 


328509 


8-3066239 


4-101.5661 


•014492754 


70 


4900 


343U00 


8-3666003 


4-121-2853 


•01428.5714 


7J 


5041 


357911 


8-4261408 


4- 1408 J 78 


•014084517 


72 


5 J 84 


373248 


8-4852814 


4-160](,76 


-013888889 


73 


5329 


389017 


8-544()():{7 


4-1793390 


•013698630 


74 


5476 


405224 


8-6023253 


4-19d3j64 


•013513514 


75 


5625 


421875 


8-6602.J40 


4-2171633 


•013333333 


76 


5776 


438976 


8-7177979 


4-2358236 


•0131.57895 


77 


5929 


456533 


8-7749644 


4-2543210 


-012987013 


78 


6084 


474552 


8-8317609 


4-2726586 


-01-2820513 


79 


6-241 


493039 


8-8881944 


4-2908404 


-01-26.58228 


80 


6400 


512000 


8-9442719 


4-3088695 


•012500000 


81 


6561 


531441 


90000000 


4-3-267487 


•012345679 


82 


6724 


551 368 


90553851 


4-3444815 


012195122 


83 


6889 


571787 


9-11043.36 


4-3620707 


•012048193 


84 


7056 


592704 


9-1651514 


4-3795191 


•011904762 


85 


7225 


614125 


9-219.5445 


4-3968296 


011764706 


86 


7396 


636056 


9-2736 185 


4-4140049 


•011627907 


87 


7569 


658503 


9-3273791 


4-4310476 


-011494253 


88 


7;44 


681472 


9-3808315 


4-4479602 


•011363636 


89 


7921 


704969 


9-4339811 


4-4647451 


•011235955 


90 


8100 


729000 


94868330 


4-4814047 


■011111111 


91 


8281 


753571 


9-5393920 


4-4979414 


•010989011 


92 


8464 


778688 


9-5916630 


45143574 


-010869565 


93 


8649 


804357 


9-6436508 


4-5306549 


0107.52688 


94 


8836 


830584 


9-6953597 


4-5468359 


•010638298 


95 


9025 


857375 


9-7467943 


4-5629026 


-010.526316 


96 


9216 


884736 


9-7979590 


4-5788570 


-010416667 


97 


9409 


912673 


9-8488578 


4-5947009 


•010309278 


98 


9604 


941192 


9-8994949 


4-6104363 


•010204082 


99 


9801 


970299 


9-9498744 


4-6-260650 


-010101010 


100 


10000 


1000000 


100000(100 


4-641.5888 


010000000 


lOJ 


10201 


1030301 


100498756 


4-6570095 


•009900990 


102 


10404 


1061208 


10-0995049 


4-6723287 


-009803922 


103 


10609 


1092727 


10-1488916 


4-6875482 


-009708738 


104 


10816 


1124864 


101980390 


4-7026694 


009615385 


105 


11025 


1157625 


10-2469508 


4-7176940 


•009523810 


106 


11236 


1191016 


10-2956301 


4-7326235 


-009433962 


107 


11449 


1225043 


l>/-3440804 


4-7474594 


•009345794 


108 


11064 


1259712 


10-3923048 


4-7622032 


•009259259 


109 


llddl 


1295029 


10-4403065 


4-7768562 


009174312 


110 


12J00 


1331000 


10-4880885 


4-7914199 


•009090909 


111 


12321 


1367631 


10-5356.538 


4-8058995 


•009009009 


m 


I'^tl 


}mw 


|g-58300§5} 


4-820^845 


•U0892857J 



164 



SQUARES, CUBES, ETC., OF NUMBERS, 



No. 


Squares. 


Cube*. 


Square Roots. 


Cube Roots. 


Reciprocal* 


113 


12769 


1442897 


10-6301458 


4-8345881 


•008849.558 


114 


12996 


1481.544 


10-6770783 


4-8488076 


•008771930 


115 


1.3225 


1.520875 


10-7238053 


4-8629442 


•0086956.52 


116 


13456 


1560896 


10-7703296 


4-8769990 


•008620690 


117 


1.3689 


1601613 


10-8166538 


4-8909732 


•008547009 


118 


13924 


1643032 


10-8627805 


4-9048681 


•008474576 


119 


14161 


16851.59 


10-9087121 


4-9186847 


•008403361 


120 


14400 


1728000 


10-9544512 


4-9324242 


•008333333 


121 


14641 


1771561 


11-0000000 


4-9460874 


•008264463 


122 


14834 


1815848 


11-0453610 


4-9596757 


-008196721 


123 


15129 


1860867 


n -090.5365 


4-9731898 


-0081,30081 


124 


1.5376 


1906624 


11-13.5.5287 


4-9866310 


-008064516 


125 


15625 


1953125 


11-1803399 


5-0000000 


-008000000 


126 


15876 


2000376 


11-2249722 


.50132979 


-007936508 


127 


16129 


2048383 


11-2694277 


5-0265257 


•007874016 


128 


16384 


20971.52 


11-3137085 


5-0396842 


•007812500 


129 


16641 


2146689 


11-3.578167 


50527743 


-007751938 


130 


16900 


2197000 


11-4017.543 


50657970 


-007692303 


131 


17J61 


2248091 


11-445.5231 


50787531 


•007633588 


132 


17424 


2299968 


11-4891253 


5-0916434 


•007575758 


133 


17689 


23.52637 


11-5325626 


5- J 044687 


-007518797 


134 


179.56 


2406104 


11 -57.58369 


51172299 


•007462687 


135 


18225 


2460375 


11-6189500 


5-1299278 


•007407407 


136 


18496 


251.5456 


11-6619038 


.5-1425632 


•007.3.52941 


137 


18769 


2.571353 


11-7046999 


5-1551.367 


•007299270 


138 


19044 


2828072 


11-7473401 


5-1676493 


■007246377 


139 


19321 


2685619 


11-7898261 


5-1801015 


•007194245 


140 


19600 


2744000 


11-8321596 


5-1924941 


•007142857 


141 


19881 


280322] 


11-8743421 


5-2048279 


•007092199 


142 


20164 


2863288 


11-91637.53 


.5-2171034 


•007042254 


143 


20449 


2924207 


]p<)58-2607 


5-2293215 


•006993007 


144 


20736 


2985984 


]2-(t0O00OO 


5-2414828 


•006944444 


145 


21025 


3048G25 


1C;04 15046 


5-2535879 


-00(5896552 


146 


21316 


311213(5 


1208304()0 


5-2()56374 


■006849315 


147 


21609 


3176523 


12-1243.557 


5-2776321 


•006802721 


148 


21904 


3241792 


12-16.55251 


5-2895725 


•((06756757 


149 


22201 


3307949 


12-2065556 


5-.30 14.592 


•00671 1409 


150 


22500 


3375000 


12-2474487 


5-3132928 


-00666()6(i7 


151 


22801 


3442951 


12-2882057 


5-3250740 


-006622517 


152 


23104 


3511808 


12-3288280 


5-3368033 


-006.578947 


153 


23409 


3581577 


12-3693169 


5-3484812 


•006535948 


154 


23716 


3652264 


12-4096736 


5-3601084 


•006493506 


155 


24025 


3723875 


12-4498996 


5 37168.54 


•006451613 


156 


24336 


3796-1 ]fi 


12-4809960 


5-3832126 


•006410256 


157 


24649 


3869893 


12-5299641 


5-3946907 


•006369427 


158 


24964 


3944312 


12 5698051 


5-4061202 


•006329114 


159 


25281 


4019G79 


12-6095202 


5-4175015 


•006-289308 


160 


25G00 


4096000 


12 6491108 


5-4288352 


-0062.50000 


161 


25921 


4173281 


12-6885775 


5-4401218 


•00(5211180 


162 


26244 


4251.528 


12-7279221 


5-4513618 


•006172840 


163 


26569 


4330747 


12-7671453 


5-4625.556 


•006134969 


164 


26896 


4410944 


12-8062485 


5-4737037 


•006097561 


165 


27225 


4492125 


12-84.52326 


5-4848066 


•0060(50606 


166 


27556 


4574296 


12-8840987 


5-4958647 


•006024096 


167 


27889 


4657463 


12-9228480 


5-5068784 


•005988024 


168 


28224 


4741632 


129614814 


5-5178484 


•005952.381 


169 


28561 


4826809 


13-0000000 


5-5287748 


•005917160 


170 


28900 


4913000 


13-0384048 


5-5396583 


•005882353 


171 


29241 


5000211 


13 0766968 


5-5504991 


•005847953 


172 


29584 


5088448 


13 1148770 


5-5612978 


•005813953 


173 


29929 


5177717 


13-1529464 


5-5720546 


•005780347 


174 


30276 


5268024 


13-1909060 


5-5827702 


•005747126 



SQUARES, CUBES, ETC, OF NUMBERS. 



165 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals, 


175 


.30625 


5359375 


13-2287566 


5-5934447 


005714286 


176 


30976 


5451776 


13-2664992 


5-6040787 


•00.5681818 


177 


31329 


5545233 


13-.3041347 


.5-6146724 


•005649718 


178 


31684 


5639752 


13 3416641 


5-62.52263 


•005617978 


179 


32041 


5735339 


13-3790882 


5 63.57408 


•005586592 


180 


32400 


5832000 


13-4164079 


5-6462162 


•005555556 


181 


32761 


5929741 


13-4536240 


5-6566528 


•005.524862 


182 


33124 


6028568 


13-4907376 


.5-6670511 


•005491.505 


183 


33489 


6128487 


13-5277493 


.5-6774114 


•00546448 1 


184 


33856 


6229504 


l3-564(i600 


5-6877340 


•005434783 


185 


34225 


6331625 


13-6014705 


5-6980192 


•00540.5405 


186 


34596 


6434856 


13-6381817 


5-7082675 


•005376344 


187 


34969 


6539203 


13-6747943 


5-7184791 


•005347594 


188 


35344 


6644672 


]3-7!13U92 


5-7286.543 


•005319149 


189 


35721 


6751269 


13-7477271 


5-7387936 


•005291005 


UK) 


36100 


6859000 


13-7840488 


5-7488971 


•00.5263158 


191 


36481 


6967871 


13-8202750 


5-7589652 


•005235602 


192 


36864 


7077888 


13-8564065 


5-7 -89982 


•00.5208333 


193 


37249 


7189017 


13-8924440 


5-7789966 


•005181347 


194 


37636 


73013&4 


1:^-9283883 


5-7889604 


•005154639 


195 


38025 


741 4875 


] 3964-2400 


5-7988900 


•005128205 


196 


38416 


7529536 


14-00001)00 


5-8087857 


•005102041 


197 


38809 


7645373 


14-0356688 


5-8186479 


-005076142 


196 


39204 


7762392 


14071^:173 


5-8284767 


•005050.505 


199 


39601 


7880599 


1410G7360 


5-8382725 


•005025126 


200 


40000 


8000000 


14-14213.-).'> 


5 84803.55 


•005000000 


201 


40401 


8120601 


14- 1774469 


5-8.577660 


•004975124 


202 


40804 


8242408 


14-2i207O4 


5-8674643 


. -004950495 


203 


41209 


8365427 


14-24T8(;68 


.5-8771307 


•004926108 


204 


41616 


8489664 


14-28-2H569 


5-8867653 


•004901961 


205 


42025 


8615125 


14-3178-21 1 


5-8963685 


•004878049 


206 


42436 


8741816 


14-3527001 


59059406 


•00485436'.) 


207 


42849 


8869743 


14.3874946 


.5-91.54817 


•004830918 


208 


43264 


8998912 


14-4222051 


5-9249921 


-004807692 


209 


43681 


9129329 


14-4568.323 


5-9344721 


•004784689 


210 


44100 


9261000 


14-4913767 


5-9439220 


-004761905 


211 


44521 


9393931 


14-5258390 


.5-9.5.33418 


•C04739336 


212 


44944 


9528128 


14-5602198 


5-9627320 


•004716981 


213 


45369 


9063597 


14-5945195 


5-9720926 


•004r;94836 


214 


45796 


9800344 


14-6287388 . 


5-9814240 


•004672897 


215 


46225 


9938375 


14-6628783 


5-9907264 


•004651 163 


216 


46056 


10077696 


14-6909385 


GOOOOOOO 


-004629630 


217 


47089 


10218313 


147309199 


6-0092450 


-004608295 


218 


47524 


10360232 


14-7648231 


6-0184617 


•004.587156 


219 


47961 


10503459 


14-7986486 


6-0276502 


•004566210 


220 


48400 


10648000 


14-8323970 


6-0368107 


■004.5454.55 


221 


48841 


10793861 


14-8660687 


60459435 


•004524887 


222 


49284 


10941048 


14-8996644 


60550489 


•004501505 


223 


49729 


11089567 


14-9331845 


6-0641270 


•004484305 


2'J4 


50176 


11239424 


14-9006-295 


6-0731779 


•0044()4286 


OOj 


50625 


11390625 


15-.KJ00000 


6-0822020 


-004444444 


226 


51076 


11543176 


15-033-2964 


6-0911994 


-004424779 


227 


51529 


11697083 


15-0665192 


6-1001702 


-004405286 


228 


51984 


1 1852352 


15 0996689 


6-1091 147 


-004385965 


229 


.52441 


12008989 


15 1327460 


6-1180332 


-004366812 


230 


52900 


12167000 


15- 1657509 


6-12692.57 


•004347826 


231 


53.161 


12326391 


15-1986842 


6-1357924 


•004329004 


232 


53824 


12487168 


15-2315462 


6-1446337 


•004310345 


233 


54289 


12649337 


15-2643-375 


6-1534495 


•004291845 


234 


547.56 


12812904 


15-2970585 


6-1622401 


•004273504 


235 


55225 


12977875 


15-3297097 


61710058 


•0042.5.5319 


236 


55696 


13144256 


15-3622915 


6-1797466 


-00423V288 



166 



SQUARES, CUBES, ETC., OF NmfBERS. 



Na 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciproca!*. 


237 


56169 


13312053 


15-3948043 


6-1884G28 


•004219409 


238 


56644 


13481272 


15-4272486 


6-1971544 


•004-201681 


239 


57121 


13651919 


15-4596248 


6-20.58218 


•004184100 


240 


57600 


13824000 


15-49] 9334 


6-2144650 


•004166667 


24) 


58081 


13997521 


15-5241747 


6-2230843 


•004 14:4378 


212 


585G4 


14172488 


15-5563492 


6-2316797 


•004 13223 J 


243 


59049 


14348907 


15-5884573 


6-2402515 


-004]1.>2-2G 


244 


59536 


14526784 


15-6204994 


6-2487998 


-004098361 


245 


60025 


14706125 


15-65-24758 


6-2573248 


-004081633 


246 


60516 


14886936 


15-6843871 


6-26.58266 


•004065041 


247 


61009 


15069223 


157162336 


6-2743054 


-004048583 


248 


61504 


15252992 


15-7480157 


6-2827G13 


-004(«2-2.")8 


249 


62001 


15438249 


15-7797338 


6-2911946 


-004016064 


250 


6-2500 


15625000 


15-8113883 


6-29960.53 


•0O4U00000 


251 


63001 


15813251 


15-8429795 


6-3079935 


•003984064 


252 


63504 


16003008 


15-8745079 


6-3163536 


-003968254 


253 


64009 


16194-277 


15-9059737 


6-3247035 


-003952569 


254 


64516 


163870(>4 


15-9373775 


6-3330256 


-003937008 


255 


65025 


16581375 


15-9687194 


6-3413257 


•003921589 


256 


65536 


16777216 


16-0000000 


6-3496042 


-00390G250 


257 


68049 


lGiJ74593 


lG-03 12195 


6-3.57861 1 


•00:i8!H051 


258 


665G4 


17173512 


1G0G23784 


6-3660968 


•0038759G9 


259 


67081 


17373979 


16-0934769 


6.37431 J 1 


-003861004 


260 


67600 


17576000 


IG- 1245 155 


6-3825043 


-003846154 


261 


68121 


17779581 


16-1554944 


6-39067G5 


-003831418 


262 


68644 


17984728 


16- 1864 141 


6-3988279 


•00381()794 


263 


69169 


181 91447 


16 2172747 


6-4()G9585 


•003802281 


264 


69696 


18399744 


10-2480768 


6-4150687 


•003787879 


265 


70225 


18609625 


16-2788206 


6-4231583 


•003773585 


266 


70756 


18821096 


16-3095064 


6-4319276 


•003759398 


267 


71289 


19034163 


16-3401346 


6-4399767 


•003745318 


268 


71824 


19248832 


16-3707055 


6-4473057 


-003731343 


269 


72361 


19465109 


16-4012195 


6-4553148 


•003717472 


270 


72900 


19683000 


16-4316767 


6-4633041 


•003703704 


271 


73441 


19902511 


16-4620776 


6-4712736 


•003690037 


272 


73984 


2U123648 


16-4924225 


6-479-2236 


•003676471 


273 


74529 


20346417 


16-5227116 


6-4871541 


•003663004 


274 


75076 


20570824 


16-5529454 


6-4950653 


•003649635 


275 


75625 


20796875 


16-5831240 


6-5029572 


-00363(5364 


276 


76176 


21024576 


16-6132477 


6-5108300 


•003623188 


277 


76729 


21253933 


16-6433170 


6-5186839 


•003610108 


278 


77284 


21484952 


16-673332) 


6-5265189 


•003597122 


279 


77841 


21717639 


16-7032931 


6-5343351 


•00.3.584229 


280 


78400 


2195-20U0 


16-7332005 


6-54213-26 


•003.571429 


281 


78961 


22188041 


16-7630546 


6-5499] 16 


-003.5.58719 


282 


79524 


2-2425768 


16-7928556 


6-5576722 


•00354(5099 


283 


80089 


226G5187 


16-8226038 


6-5654144 


-003533569 


284 


80656 


22906304 


16-8522995 


6-5731385 


-003.522127 


285 


81225 


23149125 


16-8819430 


6-5808443 


•00350H772 


286 


81796 


23393G56 


16-9115345 


6-588.5323 


•003496503 


287 


82369 


23639903 


16-9410743 


6-5962023 


•003484321 


288 


82944 


23887872 


16-9705627 


6-6038545 


•003472222 


289 


83521 


24137569 


17-0000000 


6-6114890 


•00346020H 


290 


84100 


24389000 


17-0293864 


6-6191060 


•003448276 


291 


84681 


24642171 


17-0587221 


6-62(57054 


-0034.3643) 


292 


85264 


24897088 


17-0880075 


6-6342874 


-003424f55S 


293 


85849 


25153757 


171172428 


6-6418522 


•0034129'!:) 


294 


86436 


25412184 


17-1464282 


6-6493998 


•003401 3^51 


295 


87025 


25672375 


17-1755640 


6-65()9302 


•003.38,K{| 


296 


87616 


25934336 


17-2046505 


6-6644437 


•00.3378378 


297 


88209 


26198073 


17-2336879 


6-6719403 


•003367003 


298 


88804 


26463592 


17-2626765 


6-6794200 


-0033o5705 



SQUARES, CUBES. ETC., OF NUMBERS. 



107 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


239 


8M01 


2G730899 


17-291GJ65 


6-6868831 


•003344482 


300 


9;)noo 


27000000 


1 7-3205081 


6-6943295 


•00333:i;i33 


301 


90001 


27270901 


17-3493516 


6-7017593 


•003322259 


30'i 


91204 


27543608 


17-3781472 


6-7091729 


•0033112.58 


303 


91809 


27818127 


17-4068952 


6-7165700 


•003301330 


304 


924 J 6 


28094464 


17-4355958 


6-7239508 


•003289474 


305 


G3;;25 


28372025 


17-4042492 


6-7313155 


•003278G89 


306 


93G36 


28652616 


17-4928557 


6-7386641 


•0032G7974 


307 


94249 


28934443 


17-5214155 


6-74599G7 


•0032.57329 


338 


94804 


29218112 


17-5439288 


6-7533134 


•003240753 


309 


93481 


29503G29 


17-5783958 


6-760G143 


•00323G24G 


310 


9G100 


29791000 


17-0068169 


6-76789J5 


•0032258.;G 


311 


9G721 


3008C23J 


17-6351921 


6-7751 G90 


•003215434 


312 


97344 


30371328 


17-6035217 


6-7824229 


•003205128 


3J3 


97969 


30G64297 


17-G9180G0 


6-7896613 


•003194888 


3J4 


98596 


30959144 


17-7200451 


6 7968844 


•003184713 


3J5 


99225 


31255875 


17-7482393 


6-8040921 


•003174;;^3 


3J6 


99856 


31554496 


17-7763883 


6-8112847 


•0031G4.:57 


317 


100489 


31855013 


17-8044938 


6-8184620 


•003154574 


318 


101124 


32157432 


17-8325545 


6-825G242 


•003144G.54 


319 


1017G1 


32461759 


17-0605711 


6-8:]27714 


•003134796 


320 


102400 


32768000 


17-8885438 


6-8399037 


•003125.;30 


321 


103041 


33076161 


17-9164729 


6-84702 J 3 


•003115205 


322 


103684 


33386248 


17-9443584 


6-85412-10 


•0031055;)J 


323 


104329 


33698267 


17-9722008 


G-8G12120 


■0O3O95.)75 


324 


104976 


34012224 


18-00000:;0 


6-81382855 


•00308G420 


325 


105625 


34328125 


18-02775G4 


6-87534 W 


•00307G923 


326 


106276 


34645976 


18-0554701 


6-8823888 


0030G7485 


327 


106929 


349G5783 


18-0831413 


6-8894188 


•003058104 


328 


107584 


35287552 


18-1107703 


6-8964345 


•003048780 


329 


108241 


35611289 


181383571 


6-9034359 


•003039514 


330 


108900 


35937000 


18-1659021 


6-9104232 


•003030303 


331 


109561 


36264691 


18-1934054 


6-9173964 


•003021148 


332 


110224 


36594368 


18-2208G72 


G-9243556 


•003012048 


333 


110889 


36926037 


•18-2482376 


6-9313008 


•003003003 


334 


111556 


37259704 


18-2756G69 


6-9382321 


•002994012 


335 


112225 


37595375 


18-3030052 


6-9451496 


•002985075 


336 


112896 


37933056 


18-3303028 


6-D520533 


•002976190 


337 


113569 


38272753 


18-3575598 


6-95Hr)434 


•002967359 


338 


114244 


38614472 


18-3847763 


6-9058198 


•0029.58580 


339 


114921 


38958219 


18-4119526 


6-9720826 


■002949853 


340 


115600 


39304000 


18-4390889 


6-9795321 


•002941176 


341 


116281 


39G51821 


18 4661853 


6-9863681 


-002932551 


342 


1169G4 


40001688 


18-4932420 


6-9931906 


•002923977 


343 


117649 


40353607 


1^^5202592 


7-0000000 


-002915452 


344 


118336 


40707584 


18-5472370 


7-00G7962 


-002906977 


345 


119025 


41063625 


18-5741756 


7-0135791 


-002898551 


346 


119716 


41421736 


18-6010752 


7-0203490 


•002890173 


347 


120409 


4178 J 923 


18-6279360 


7-0271058 


•002881844 


348 


121104 


42144192 


18-6547581 


7-0338497 


•002873563 


349 


121801 


42508549 


18-6815417 


7-U4058015 


-00286.53.30 


350 


122500 


42875000 


18-7082869 


7-0472987 


•002857143 


351 


123201 


43243551 


18-7349940 


7-0540041 


•002849003 


352 


123904 


43614208 


18-7616630 


7-0606967 


•002840909 


353 


124609 


43986977 


18-7882942 


7-0673767 


•002832861 


354 


125316 


44361864 


18-8148877 


7-0740440 


■002824859 


355 


126025 


44738875 


18-8414437 


7-0806988 


•002816901 


356 


126736 


45118016 


18-8679623 


70873411 


•002808989 


357 


127449 


45499293 


18-8944436 


7-0939709 


•002801120 


358 


128164 


45882712 


18-9208879 


7-1005885 


•002793296 


359 


128881 


46268279 


18-947W53 


7-1071937 


•0027855J5 


360 


129«(H> 


46656000 


18-9736660 


71137866 i 


•002777778 



168 



SQUARES, CCBES, ETC., OF NUMBERS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. Reciprocals. 


361 


130321 


47045881 


190000000 


7-1203674 


002770083 


362 


131044 


47437928 


190262976 


7-1269360 


002762431 


363 


131769 


47832147 


190.525589 


71334925 


002754821 


364 


132496 


48228r)44 


190787840 


71400370 


002747253 


365 


133225 


4862712.5 


191049732 


7 1465695 


002739720 


366 


133956 


49027896 


191311265 


7-15.30901 


00273224) 


367 


134689 


49430863 


191572441 


71.59.5988 


0027247(t6 


368 


135424 


49836032 


191833261 


7-1660957 


00271731)1 


369 


13G161 


50243409 


192093727 


7 1725809 


002710(W7 


370 


136900 


50653000 


19 23.53841 


7-1790.544 


002702703 


371 


137041 


5 10648 U 


192613603 


7-18.55162 


002695418 


372 


138384 


51478848 


19 2873015 


7-1919663 


002688172 


373 


139129 


51895117 


193132079 


7-1984050 


002680965 


374 


139876 


52313624 


193390796 


7 2048322 


002673797 


375 


140625 


52734375 


W3649167 


7-2112479 


002666667 


376 


141376 


53157376 


193907194 


7-2176522 


002659574 


377 


142129 


53582633 


19 4164878 


722404.50 


00265252J 


378 


142884 


540101.32 


1944229,91 


7-2304268 


002645503 


379 


143641 


54439939 


194679223 


7 23679/2 


002638521 


380 


144400 


M872000 


194935887 


7 2431565 


002631579 


3«1 


145181 


55306341 


19 5192213 


72495045 


002024672 


382 


145924 


55742968 


19 5448203 


7 2558415 


002617801 


383 


146689 


5G181887 


195703858 


72021675 


OOQGlOlKJo 


384 


147456 


56G23104 


19 5959179 


72684824 


002G041G7 


385 


148225 


57066625 


19 6214 169 


7-2747864 


002597403 


386 


148996 


57512456 


19 6468827 


7-2810794 


002590674 


387 


149769 


57960603 


19 6723156 


7-2873617 


002583979 


388 


150544 


58411072 


19 6977156 


7-2936330 


002577320 


389 


151321 


58863869 


197230829 


7 2998936 


00257(;()94 


390 


152100 


59319000 


197484177 


7-3061436 


002564103 


31M 


152881 


59776471 


19 7737199 


7-312.3828 


002557.545 


W'.Vl 


15361)4 


60236288 


19 7989899 


7-318()lI4 


00255 IOC J 


393 


154449 


60698457 


19 8242276 


7-32482;)5 


002544.529 


394 


155236 


61162984 


19 8494332 


7 3310369 


0!)2.538;)7l 


395 


156025 


61629875 


19 8746069 


7-3372339 


002.531646 


396 


156816 


62099136 


19-8997487 


7-3434205 


002525253 


397 


157609 


62570773 


19-9248588 


7-3495966 


002518892 


398 


158404 


63044792 


19 9499373 


7-3557624 


002512563 


L99 


159201 


63521199 


19 9749844 


7-3619178 


002.506266 


400 


160000 


64000000 


200000000 


7-3(i8t)630 


002.500000 


401 


160801 


64481201 


20 0249844 


7-3741979 


002493766 


402 


161604 


64964808 


20-0499377 


7-3803227 


002487.562 


403 


162409 


65450827 


20 0748599 


7-3864373 


002481390 


404 


163216 


65939264 


20 0997512 


7-3925418 


002475^248 


405 


164025 


66430125 


20 1246118 


7 3986363 


002469136 


406 


164836 


66923416 


20-1494417 


7-41)47206 


00241)3054 


407 


1(35649 


67419143 


20 1742410 


74J079.50 


002457002 


408 


1664()4 


67917312 


20-1990099 


7-4168595 


002450980 


409 


167281 


68417929 


20 2237484 


74229142 


002444988 


410 


168100 


68921000 


20 2484.567 


7-4289589 


002439024 


411 


168921 


69426531 


20 2731.349 


7 4349938 


1102433090 


412 


1(59744 


ti; (934528 


20 2!>77831 


7 4410189 


002427184 


413 


170569 


70444997 


203224014 


7 4470342 


()024213(,8 


414 


171396 


70957944 


20 3469899 


74530399 


0024 J 54,59 


415 


172225 


71473375 


20-3715488 


7 4590359 


002409639 


416 


173056 


71991296 


20-.3960781 


7-4650223 


002403846 


417 


173889 


72511713 


20-4205779 


7 4709991 


002398()82 


418 


174724 


73034632 


20-4450483 


7-4769664 


002392344 


419 


175561 


73560059 


20-4694895 


7-4829242 


002386635 


420 


176400 


74088000 


20-4939015 


7-4888724 


002380952 


421 


177241 


74618461 


20-5182845 


7-4948113 


00237.5297 


422 


178084 


75151448 


20v42ti386 


7-3007406 


0023696G8 



SQUARES, CUBES, ETC., OF NUMBERS. 



169 



N 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


423 


178929 


75686967 


20 1669038 


75066607 


-002.364006 


424 


179776 


76225024 


20-5912003 


7-512.5715 


-002358491 


4'i5 


180625 


76765625 


20-615,1281 


7-51847.30 


002352941 


4'J6 


181476 


77308776 


20-0397674 


7.1243652 


•002347418 


4'27 


182329 


778.14483 


20-6639783 


7-5302482 


•002341920 


4-J8 


18^184 


784027.12 


20-6881609 


7-5361221 


■002330449 


429 


184041 


78953589 


20-71231.12 


7-5419867 


002.331002 


4;<l) 


184900 


79507000 


20-7364414 


7-5478423 


•002325581 


431 


185761 


80062991 


207605395 


7-5536888 


002.3-20186 


4:^2 


186624 


80621568 


20-7846097 


7-5595263 


002314815 


43:{ 


187489 


81182737 


20-8086520 


7-5653548 


-002309469 


434 


188356 


81746504 


20-8326667 


7 5711743 


•002.304147 


43,-) 


189225 


82312875 


20-8566536 


7-5709849 


•002298851 


43C. 


190096 


82881856 


20-8806139 


7-5827865 


-002293.178 


437 


190969 


83453453 


20 904,1450 


7-5885793 


•002288330 


438 


191844 


84027672 


20-9284495 


7-5943633 


-002283105 


43M 


192721 


84604519 


20-9523268 


7-6001385 


-002277904 


440 


193600 


85184000 


209761770 


7-6059049 


-002272727 


441 


194481 


85766121 


21-0000000 


7-6116626 


-002267574 


44-2 


19.1364 


86350888 


210237960 


7-6174116 


-002262443 


443 


196249 


86938307 


210475652 


7-6231519 


•002257.336 


444 


197136 


87528384 


210713075 


7-6288837 


•002252252 


44.1 


198025 


88121125 


210950231 


7-6346067 


-002247191 


44G 


198916 


88710536 


211187121 


7-6403213 


002242152 


447 


199809 


89314023 


21-1423745 


7-6460272 


-002237136 


448 


2,00704 


89915392 


21-1660105 


7-6517247 


-0022.32143 


443 


201601 


90518849 


21-1896201 


7-6.174138 


002227171 


450 


202500 


9112.5000 


21 2132034 


7-6630943 


-002222222 


4.11 


203401 


91733851 


212367606 


7-6087065 


002217295 


452 


204C04 


92345408 


21-2602916 


76744303 


002212389 


433 


205209 


92959677 


21-2837967 


7-6800857 


•002207506 


4.14 


206116 


93576664 


21-3072758 


7-6857328 


•002202643 


4.1.1 


207025 


94196375 


21-3307290 


7-691.3717 


•002197802 


456 


207936 


94818816 


21-3541565 


7-6970023 


-002192982 


457 


208849 


95443993 


21-3775583 


7 7020246 


-002188184 


458 


209764 


96071912 


21-4009346 


7-7082388 


-002183406 


4.19 


210081 


96702579 


214242853 


7-7188448 


-002178649 


4C0 


211 600 


97336000 


21-4476106 


7-7194426 


•00217.3913 


4r,i 


212521 


97972181 


21-4709106 


7 7250325 


-002169197 


4G2 


213444 


98611128 


21-4941853 


77306141 


•002164502 


4G3 


214309 


99252847 


21-5174348 


7-7361877 


•002159827 


4G4 


21529G 


99897344 


21-540r5-:.2 


7-7417532 


•0021.15172 


4C5 


21G225 


100.144625 


21-5038.187 


7-7473109 


■002150538 


406 


217156 


101194696 


21-5870331 


7-7528606 


•002145923 


467 


218089 


101847563 


21-6101828 


7-7584023 


•002141328 


468 


219024 


102503232 


21-6333077 


7-7639361 


002136752 


460 


219961 


103161709 


21-6564078 


7-7694620 


002132196 


470 


22J900 


103823000 


21-6704834 


7-7749801 


002127660 


471 


221841 


104487111 


21-7025344 


7-7804904 


•002123142 


472 


222784 


1051.14048 


21-72.1.1610 


7-7859928 


002118044 


473 


223729 


10.182 i81 7 


21 7485'-32 


77914875 


002114165 


474 


. 224676 


106496424 


217715411 


7-7969745 


002109705 


475 


225625 


107171875 


21 7944947 


7-8024538 


002105263 


4T6 


226576 


107850176 


218174242 


7-8079254 


002100840 


477 


227.129 


108.131333 


21-8403297 


7-8133892 


002096486 


478 


228484 


1092153.52 


218632111 


7-81884.16 


■002092050 


479 


229441 


109902239 


21-8860086 


7-8242942 


0U2087683 


480 


230400 


110592000 


21-9089023 


7-8297353 


-002083333 


481 


231301 


111284041 


219317122 


7-8351688 


■002079002 


482 


232324 


111980168 


21-9544984 


78405949 


002074689 


483 


233289 


112678587 


21-9772610 


78460134 


-002070393 


484 


234256 


113379904 


820000000 


78514244 


-002066Jie 



no 



SQUARES, CCBES, ETC., OF NUMBERS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocal* 


4SJ 


235225 


114084125 


220227155 


7-8568281 


-0020618.56 


4J3 


2.iGl% 


11479125*! 


2204.-)4.j77 


78622242 


-002057613 


4.;7 


237169 


115501303 


- 22(){i807(i5 


7 86761.30 


-002053388 


438 


238144 


116214272 


22()907220 


7-8729944 


-002049180 


4t<d 


233121 


118930169 


221133444 


7-87838H4 


-002044990 


4J0 


2-l,)I00 


117649000 


22 1359436 


7 8837352 


-002040816 


4!»1 


241081 


118370771 


221585198 


7-8890946 


•002036660 


4. .J 


24201)4 


1190.35488 


221810730 


7-89444(i8 


-002032520 


4jJ 


2430-1^ 


119823157 


22 2038033 


7-8997917 


-002028398 


4J4 


244033 


120553784 


22 2281108 


7-90512.^4 


-002024291 


4d.-, 


245025 


121287375 


22 '2485955 


7-9104599 


-002020202 


4;M 


240016 


122023936 


22 2710575 


7-9157832 


-002016129 


4i)7 


247001) 


122763473 


22-2934968 


7-9210994 


-002012072 


4^)8 


248004 


123505992 


223159136 


7-9284085 


•002008032 


4JJ 


249001 


124251499 


22 3383079 


7-9317104 


•002004008 


5UD 


250000 


125000000 


22-3806798 


7-9370053 


-002000000 


501 


2510Ci 


125751501 


22 3830293 


7-9422931 


•001996008 


50-2 


252004 


126506008 


22-4053565 


7-9475739 


-001992032 


503 


253009 


127263527 


22-4276615 


79528477 


•001988072 


504 


254010 


128024084 


22 4499443 


7-9581144 


•001984127 


5():> 


255025 


128787825 


22 4722051 


7-9633743 


•001980198 


503 


253033 


129554216 


2249444.38 


7-9686271 


•001976285 


507 


257049 


130323843 


22 5168805 


7-9738731 


•001972387 


503 


258064 


131098512 


22 5388553 


7-9791122 


-001968504 


53.) 


259081 


131872229 


22 5810283 


7-9843444 


•001964637 


51) 


230100 


132651000 


22-5331796 


7-989^837 


•001960785 


511 


261121 


133432831 


22-6053091 


7-9947883 


•001956947 


512 


232144 


134217728 


22-6274170 


8-00C0030 


•001953125 


513 


233169 


135005697 


22-6495033 


8-00520:3 


•001949318 


514 


284196 


135798744 


22-6715381 


80104032 


•001945525 


515 


265225 


136590875 


22 0938114 


8 0155943 


•001941748 


5r, 


266256 


137383035 


22 7150331 


80207794 


•001937984 


517 


267289 


138188413 


22 737834 1 


8 0259574 


•001934236 


5 IS 


233-24 


133991832 


22-7598131 


8 0511287 


•0019.30502 


5i;) 


239331 


139798359 


22-7815715 


80362935 


•001926782 


520 


270400 


149308000 


22-8035335 


80414515 


■001923077 


521 


271411 


141420761 


228254244 


8-0466030 


■001919386 


522 


272484 


142238648 


22-8473193 


80517479 


•091915709 


523 


273529 


143055667 


228691933 


8-0568832 


001912046 


524 


274576 


1438/7824 


228910-183 


80620130 


•001908397 


525 


275625 


144703125 


22 9123785 


8-0671432 


-001904762 


523 


276676 


145531576 


22 9",43399 


8 0722323 


•001901141 


527 


2VV729 


143133183 


229564836 


8 0773743 


-001897533 


523 


278784 


M7 197932 


22 9732536 


80824303 


•001893939 


52J 


279841 


143335333 


23 0000000 


80875734 


-001890359 


53) 


23.9JD 


148877001 


230217289 


80928723 


•001886792 


5! I 


281931 


149721231 


23 0434372 


80977539 


-001883239 


5:j 


283324 


153588783 


23C651252 


8 1023333 


•001879699 


5:3 


2.84089 


151419437 


23 0807928 


8-1079128 


•001876173 


531 


285153 


152273334 


23-1084409 


8-1129803 


•001872859 


5.15 


283225 


153133.375 


23 1303870 


8-1180414 


•0018691.'-.0 


535 


287298 


153393358 


23 15187.38 


8-1230932 


•001865672 


537 


238339 


154854153 


23-1732305 


81281447 


-001362197 


533 


289444 


155723872 


23 134^273 


8-1331870 


-0fll8.'')873fi 


539 


290521 


155590819 


23 2133735 


8-1-S2230 


■001855238 


54'J 


291600 


1 "7484000 


23 2379331 


8 1432529 


-001851852 


541 


292681 


1:8340421 


23 2594087 


8 1482765 


-001848429 


542 


293764 


159220088 


23 2338335 


8- 1532939 


•00184.W18 


543 


294849 


160103007 


23 3023604 


81.5H3051 


•001841621 


544 


235933 


160989184 


23-n238376 


8 1333102 


-0018382.35 


545 


297025 


161878625 


23 .3452351 


8- 1683002 


-00]H34H(1'-J 


iM6 


2981 13 


182771336 


233666429 


8-1733020 


mmm 



SQCAIIFS, CUBES, ETC., OF NUMBERS. 



171 



Sc.uares. 



Square Roots. Cube Roots, 



Reciprocals. 



299-209 

300304 

301401 

302500 

303601 

304704 

305809 

306916 

308025 

309 J 36 

310249 

311364 

312481 

313600 

314721 

315844 

316969 

318096 

319225 

320356 

321489 

322624 

323761 

324900 

326041 

327184 

328329 

329476 

330625 

331776 

332929 

334084 

335241 

336400 

337561 

338724 

339889 

341056 

342225 

343396 

344569 

345744 

346921 

348100 

349281 

350464 

351649 

352836 

354025 

355216 

356409 

357604 

35880J 

360U0O 

36J201 

3(32404 

363609 

364816 

366025 

367236 

368449' 

369664 



163667323 

184566592 

16546914!> 

166375000 

]672.-^4J51 

168196608 

169112377 

170031464 

170933875 

171879616 

172808693 

J 73741 IJ 2 

174676879 

175616000 

176558481 

177504328 

178453547 

179406144 

180362125 

181321496 

182284263 

183250432 

184220009 

185193000 

186169411 

187149248 

188132517 

189119224 

190109375 

191102976 

192100033 

193100552 

194104539 

195112000 

196122941 

197137368 

198155287 

199176704 

200201625 

201230056 

202262003 

203297472 

204336469 

205379000 

206425071 

207474688 

208527857 

209584584 

210644875 

211708736 

212776173 

213847192 

214921799 

216000000 

217081801 

218167208 

2h)256227 

2-2():i4dr64 

2.; 1445 J 25 

22-2545016 

223648543 

224755712 



23-3880311 

23-4093998 

23-4307490 

23-452()788 

23-4733892 

23-4946802 

23-5159520 

23-5372046 

23-5584380 

23-5796522 

23-6008474 

23-6220236 

23-6431808 

23-6643191 

23-6854386 

23-7065392 

23-7276210 

23-7486842 

23-7697286 

23-7907545 

23-8117618 

23-8327506 

23-8537209 

23-8746728 

23-8956063 

23-9165215 

23-9374184 

23-9582971 

23-9791576 

24-0000000 

24-0208243 

240416306 

24-0624188 

24-0831891 

24-1039416 

24-1246762 

24-1453929 

24-1660919 

24-1867732 

24-2074369 

24-2280829 

24-2487113 

24-2693222 

24-2899156 

24-3104996 

24-3310501 

24-3515913 

24-3721152 

24-3926218 

24-4131112 

24-4335834 

24-4540385 

24-4744765 

24-4948974 

24-5153013 

24-5356883 

24-5560583 

24-5764115 

24-5967478 

24-6170673 

24-6373700 

24-6576560 



8-1782888 
8-1832695 
8-1882441 
8-1932127 
8-1981753 
8-2031319 
8-2080825 
8-2130271 
8-2179657 
8-2-228985 
8-2278254 
8-2327463 
8-2376614 
8-2425706 
8-2474740 
8-2523715 
8-2572633 
8-2621492 
8-2670294 
8-2719039 
8-2767726 
8-2816355 
8-2864928 
8-2913444 
8-2961903 
8-3010304 
8-3058651 
8-3106941 
8-3155175 
8-3203353 
8-3251475 
8-3299542 
8-3347553 
8-3395509 
8-3443410 
8-3491256 
8-3539047 
8-3586784 
8-3634466 
8-3682095 
8-3729668 
8-3777188 
8-3824653 
8-3872085 
8-3919423 
8-3966729 
8-4013981 
8-4061 ISO 
8-4108326 
8-4155419 
8-4202460 
8-4249448 
8-4296383 
8-4343287 
8-4390098 
8-4436877 
8-4483605 
8-4530281 
8-4576906 
8-4623479 
8-4670001 
8-4716471 



•001828154 
•001824818 
•001821494 
•001818182 
•001814832 
•001811594 
•001808318 
-001805054 
•001801802 
•001798561 
•001795332 
•001792115 
•001788909 
•001785714 
•001782531 
•00 < 77935 J 
•001776199 
•001773050 
•001769912 
•001766784 
•001763668 
•001760533 
•001757469 
•001754386 
-001751313 
•001748252 
001745201 
•00J7421f.0 
•001739130 
•00173G111 
•001733102 
•001730104 
•001727116 
•001724138 
•001721170 
•001718213 
•001715266 
•0017123-29 
•00 1709402 
•001706485 
•001703578 
•001700680 
•001697793 
•001694915 
•001692047 
•001689189 
•001686341 
•001683502 
•0016806 r2 
•001577852 
•001675U12 
•001672-241 
•001669449 
•001666667 
•0016638J4 
•001661130 
•001658375 
•00 165562 J 
•00165-2893 
•001650165 
•001647446 
•001644737 



172 



SQUARES, CUBES, ETC., OF NUMBERS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciproeali). 


609 


370881 


225866529 


24-6779254 


8-4762892 


•001642036 


610 


.372100 


226981000 


24-6981781 


8-4809261 


•001639344 


611 


373321 


228099131 


24-7184142 


8-4855579 


•001636661 


612 


374544 


229220928 


24'7386338 


8-4901848 


•001633987 


613 


375769 


230346397 


24-7588368 


8-4948065 


•001631321 


614 


376996 


231475544 


24-7790234 


8-4994233 


•001628664 


615 


378225 


232608375 


24-7991935 


8-5040350 


•001626016 


616 


379456 


233744896 


24-8193473 


8-5086417 


•001623377 


617 


380689 


234885113 


24-8394847 


8-5132435 


•001620746 


618 


381924 


236029032 


24-8596058 


8-5178403 


•001618123 


619 


383161 


237176659 


24-8797106 


8-5224331 


•001615509 


m) 


384400 


238328000 


24-8997992 


8-5270189 


•001612903 


621 


385641 


239483061 


24-9198716 


8-5316009 


•001610306 


6-22 


38(3884 . 


240641848 


24-9399278 


8-5361780 


•001607717 


6-2:{ 


388129 


241804367 


24-9599679 


8-5407501 


•001605136 


624 


38:!37() 


242970624 


24-9799920 


8-5453173 


•001602564 


62.1 


390C)25 


244140625 


25-0000000 


8-5498797 


•001600000 


626 


391876 


245314376 


25-0199920 


8-5544372 


•001597444 


627 


393129 


246491883 


25-0399681 


8-5589899 


•001594896 


628 


394384 


247673152 


25-0599282 


, 8-5635377 


•001.592357 


62;l 


395641 


2488.58189 


25-0798724 


8-5680807 


•001589825 


6;iO 


39ti900 


250047000 


25-0998008 


8-5726189 


•001587302 


6:u 


398161 


251239591 


25-1197134 


8-5771523 


•001584786 


632 


399424 


252435968 


25-1396102 


8-5816809 


•001582278 


633 


400689 


253636137 


25-1594913 


8-5862047 


•001579779 


634 


401956 


254840104 


25 1793566 


8-5907238 


•001577287 


635 


403225 


256047875 


251992063 


8-5952380 


•001574803 


636 


404496 


257259456 


25-2190404 


8-5997476 


•001572327 


637 


405769 


258474853 


25-2388589 


8-6042525 


•001569859 


638 


407044 


259694072 


25 2586619 


8-6087526 


•001567398 


631) 


408321 


2609J7119 


25-2784493 


8-6132480 


•001564945 


640 


401)(i00 


262144000 


25-2982213 


8-6177388 


•001562500 


641 


410881 


263374721 


25-3179778 


8-6222248 


•001560062 


642 


412164 


264609288 


25-3377189 


8-6267063 


•001557632 


643 


413449 


265847707 


25-3574447 


8-6311830 


•001555210 


644 


414736 


267089984 


25-3771551 


8-6356551 


•001552795 


645 


416125 


258336125 


25-3968502 


8-6401226 


•001550388 


646 


417316 


26958R136 


25-4165302 


8-6445855 


•001547988 


647 


418609 


270840023 


25-4361947 


8-6490437 


•001545595 


64y 


419904 


272097792 


25-4558441 


8-6534974 


•001543210 


64<J 


421201 


273359449 


25-4754784 


8-6579465 


•001540832 


650 


■422500 


274625000 


25-4950976 


8-6623911 


•001538462 


651 


423801 


27.5894451 


25-5147016 


8-6668310 


•001536098 


652 


425104 


277167808 


25-5342907 


8-6712665 


•001533742 


653 


426409 


278445077 


25-5538647 


8-6756974 


•001531394 


654 


427716 


279726264 


25-5734237 


8-6801237 


•001529052 


655 


42'J025 


281011375 


25-5929678 


8-684.54.56 


•001.526718 


656 


430336 


282300416 


25-6124969 


8-6889630 


-001524.390 


657 


431639 


283593393 


2.5-6320112 


8-6933759 


•001522070 


658 


432964 


284890312 


2.5-6515107 


8-6977843 


•001519757 


659 


434281 


286191179 


25-6709953 


8-7021882 


•001517451 


660 


435600 


287496000 


25-69046.52 


8-7065877 


•001515152 


661 


436921 


288804781 


25-7099203 


8-7109827 


•001512859 


662 


438244 


290117528 


25-7293()07 


8-7153734 


•001510574 


663 


439569 


291 434247 


25-7487864 


8-7197596 


•001508296 


664 


440896 


2927.54944 


25-7681975 


8-7241414 


•001506024 


665 


442225 


294079625 


25-7875939 


8-7285187 


•001503759 


666 


443556 


295408296 


25-8069758 


8-7328918 


•001501502 


667 


<44889 


296740963 


25-8263431 


8-7372604 


•001499250 


668 


446224 


298077632 


25-8456960 


8-7416246 


•001497006 


669 


447561 


299418309 


25-8650343 


8-7459846 


.001494768 


^70 


4489Q0 


300763000 


25-8843582 


8-7503404 


•pOH9?537 



SQUARES, CUBES, ETC., OF KU3IBEBS. 



173 



Na 


1 Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


671 


450241 


302111711 


25-9036677 


8-7546913 


•001490313 


672 


451584 


303464448 


25-9229628 


8-7590383 


•001488095 


673 


452929 


304821217 


25-9422435 


8-7633809 


•00148.5884 


674 


454276 


306182024 


25-9615100 


8-7677192 


•001483680 


675 


455625 


307546875 


25-9807021 


8-7720532 


•001481481 


676 


456976 


308915776 


26-0000000 


8-7763830 


•001479290 


677 


458329 


310288733 


2601922.37 


8-7807084 


•001477105 


678 


459684 


311665752 


260384331 


8-7850296 


-001474920 


679 


461041 


313046839 


26-0576284 


8-7893466 


•0014727.54 


680 


462400 


314432000 


26-0768096 


8-7936.593 


•001470588 


681 


463761 


315821241 


260959767 


8-7979679 


•001468429 


682 


465124 


317214568 


26-1151297 


8-8022721 


•001466276 


683 


406489 


318611987 


26-134-2687 


8-8065722 


•0014641-2'J 


684 


467856 


320013504 


26-1533937 


8-8108681 


•001461988 


635 


469225 


3214 I 9 125 


26-1725047 


8-8151598 


•001459854 


686 


470596 


322828856 


26-1916017 


8-8194474 


•001457726 


687 


471969 


324242703 


26-2106848 


8-8237307 


•001455604 


688 


473344 


325660672 


26-2287541 


8-82SC099 


•001453488 


6b9 


474721 


327082769 


26-2488095 


8-8322850 


•001451379 


690 


476100 


328509000 


26-2678511 


8-8365559 


•001449275 


691 


477481 


329939371 


26-2868789 


8-8408227 


-001447178 


692 


478864 


331373838 


20-3058929 


8-8450854 


•001445087 


693 


480249 


332812557 


26-3248932 


8-8493440 


-001443001 


694 


481636 


334255384 


26-3438797 


8-8535985 


•001440922 


695 


483025 


335702375 


26-3628527 


8-8578489 


■001438849 


696 


484416 


337153536 


26-3818119 


8-8620952 


-001436782 


697 


485809 


338608873 


26-4007576 


8-8663375 


•0014.34720 


698 


487204 


340068392 


26-4196896 


8-8705757 


-001432665 


699 


488601 


341532099 


26-4386081 


8-8748099 


-001430615 


700 


490000 


343000000 


26-4575131 


8-8790400 


•001428571 


701 


491401 


344472101 


26-4764046 


8-8832661 


-001426534 


702 


492804 


345948408 


26-4952826 


8-8874882 


-001424501 


703 


494209 


. 347428927 


26-5141472 


8-8917063 


•001422475 


704 


495616 


348913664 


26-5329983 


8-8959204 


•001420455 


705 


497025 


350402625 


26-5518361 


8-9001304 


-001418440 


706 


498436 


351895816 


26-5706605 


8-9043366 


•001416431 


707 


499849 


353393243 


26-5894716 


8-9085387 


-001414427 


708 


501264 


354894912 


26-6082694 


8-9127369 


-001412429 


709 


502681 


356400829 


26-6270539 


8-9169311 


-00 1410437 


710 


504100 


357911000 


26-6458252 


8-9211214 


■001408451 


711 


505521 


359425431 


26-6645833 


8-9253078 


-001406470 


712 


506944 


360944128 


26-6833281 


8-9294902 


•001404494 


713 


508369 


362467097 


26-7020598 


8-9336687 


-001402525 


714 


509796 


363!I94344 


26-7207784 


8-9378433 


-001400560 


715 


511225 


365525875 


26-7394839 


8-9420140 


-001398601 


716 


512656 


367061696 


26-7581763 


8-9461809 


00139664S 


717 


514089 


368601813 


26-7768557 


8-9503438 


-001394700 


718 


515524 


370146232 


26-7955220 


8-9545029 


-001392758 


719 


516961 


371694959 


26-8141754 


8-9586581 


-001390821 


720 


518400 


373248000 


26-8328157 


8-9628095 


-001388889 


721 


519841 


374805361 


26-8514432 


8-9669570 


-001386963 


722 


521284 


376367048 


26-8700577 


8-9711007 


•001385042 


723 


522729 


377933067 


26-8886593 


8-9752406 


■001383126 


724 


524176 


379503424 


26-9072481 


8-9793766 


-001381215 


725 


525625 


381078125 


26-9258240 


8-9835089 


-001379310 


726 


527076 


382657176 


26-9443872 


8-9876373 


•001377410 


727 


528529 


384240583 


26-9629375 


8-9917620 


•001375516 


728 


529984 


385828352 


26-9814751 


8-9958899 


■001373626 


729 


531441 


38742U489 


27-0000000 


9-0000000 


-001371742 


730 


532900 


389017000 


27-0185122 


9-0041134 


-001369863 


731 


534361 


390617891 


27-0370117 


9-00822-29 


-001367989 


732 


53582 i 


3y2^3168 


27-0554985 


9-0123288 


-001366120 



174 



SQUARES, CUBES, ETC., OF NUMBERS. 



No. 


Squares. 


Cubefc 1 


Square Roots. 


Cube Roots. 


Reciprocals. 


733 


537289 


393832837 


270739727 


90164309 


■001364256 


734 


538756 


395446904 


270924344 


90205293 


•001.362398 


735 


540225 


397065375 


271108334 


9024(5239 


•001360544 


73G 


541696 


398688256 


271293199 


90287149 


•001358696 


737 


543169 


400315553 


271477439 


90328021 


•CJ1356852 


738 


544644 


401947272 


2716615.54 


9-0368857 


001355014 


739 


546121 


403583419 


271845544 


9 0409655 


•001353180 


740 1 


547600 


405224000 


27-2029410 


90450419 


•001351351 


741 


549081 


406869021 


27-22131.52 


9 0491142 


•001349.528 


742 


550564 


408518488 


27-2396769 


9-0531831 


•001347709 


743 


552049 


410172407 


27-2580203 


9 0572482 


•001345895 


744 


553536 


411830784 


27-2763634 


90613098 


•001344086 


745 


555025 


413493625 


27-2946881 


9 0033677 


•001342282 


746 


556516 


415160936 


27-3130000 


90694220 


•001340483 


747 


558009 


416832723 


27-3313007 


90734726 


•001338688 


748 


559504 


4185089D2 


27-3495887 


9 0775197 


•001336898 


749 


561001 


420189749 


27-3678644 


90815631 


•001335113 


750 


562500 


421875000 


27-3861279 


90856030 


•001333333 


751 


564001 


423564751 


27-4043792 


9-0896392 


•001331558 


752 


565504 


425259008 


27-4226184 


9-0936719 


•001329787 


753 


567009 


426957777 


27-4408455 


90977010 


•001328021 


754 


568516 


428661064 


27-4590604 


9-1017265 


•001326260 


755 


570025 


430368875 


27-4772633 


9-1057485 


•0013-24503 


756 


571536 


432081210 


27-4954542 


9-1097669 


•001322751 


757 


573049 


433798093 


27-5136330 


9-1137818 


•001321.004 


758 


574564 


435519512 


27-5317998 


91177931 


•001319261 


759 


576081 


437245479 


27 549954G 


91218010 


•001317523 


760 


577600 


438976000 


27-5680975 


9 1258053 


•00131.5789 


761 


579121 


440711081 


27-5862284 


9 1298061 


•001314060 


762 


580644 


442450728 


27 6043475 


91338034 


•001312336 


763 


582169 


444194947 


27 0224546 


91377971 


•001310616 


764 


583696 


445943744 


27-0405499 


91417874 


•001308901 


765 


585225 


447697125 


27-6586334 


91457742 


•OOJ 307190 


766 


586756 


449455096 


27-6767050 


91497.576 


•001305483 


767 


588289 


451217663 


27-6947648 


91537375 


•001303781 


768 


589824 


452984832 


27-7128129 


91577139 


•001302083 


769 


591361 


454756(509 


27-7308492 


9 1616869 


•001300390 


770 


592900 


456533000 


27-7488739 


91656565 


•001298701 


771 


594441 


458314011 


277668868 


91696225 


•001297017 


772 


595984 


460099648 


27-7848880 


91735852 


•001295337 


773 


597529 


461889917 


27 8028775 


9-1775445 


•OOI293(.61 


774 


59907(5 


463684824 


27-8208555 


91815003 


•001 29 J 990 


775 


600625 


465484375 


27 8388218 


9 1854527 


•001290323 


776 


602176 


467288576 


27-8567766 


91894018 


•001288660 


777 


603729 


469097433 


27-8747197 


9 1933474 


•001287001 


778 


605284 


470910952 


27-8926514 


9 1972897 


•001285347 


779 


606841 


472729139 


27-910.5715 


9 2012286 


•001283697 


780 


608400 


474552000 


279284801 


9 2051641 


•001282051 


781 


609961 


476379541 


27-94(53772 


92090962 


■001280410 


782 


611524 


478211768 


27-9(J42r.29 


9 2130250 


•001278772 


783 


613089 


480048687 


27 9821372 


9 2169505 


•001277139 


784 


614656 


481890304 


28 0001)000 


9 2208726 


O01275;51O 


785 


616225 


483736625 


28 0178515 


9 2247914 


•001273885 


786 


617796 


485587656 


28-03.56915 


9 2287068 


•001272265 


787 


619369 


487443403 


28-()5352()3 


9 2326189 


•001270648 


788 


620944 


489303872 


28 0713377 


9 2365277 


•001269036 


789 


622521 


491169069 


28-0891438 


9-2404333 


-001267427 


790 


624100 


493039000 


28- J 069386 


9 2443355 


•001265823 


791 


625681 


494913671 


28-1247222 


92482344 


•001264223 


792 


627624 


496793088 


28 1424946 


9 2,521300 


•001262626 


793 


628849 


498677257 


2816025.57 


92560224 


•001261034 


794 


630436 


500566184 


281780036 


9i>399U4 


001259446 



SQi^AA'ES, (JUBES, ETC., OF NUMBERS. 



175 



Sinmres. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


g:j2u->.) 


502459875 


28- 1957444 


92637973 


-001257862 


633Ul(i 


5.,4:J5.<33() 


28 2134720 


9-2676798 


-001256281 


635209 


506261573 


28-2311884 


92715592 


-001254705 


636804 


508169592 


28-2488938 


9-2754352 


001 253 J 33 


638-101 


510082399 


28 2665881 


92793081 


00J251364 


640000 


512000000 


28 2842712 


9-2831777 


-001250000 


6416U1 


513922401 


28 3019434 


9-2870444 


001248439 


643204 


515849008 


28-3196045 


92909072 


001246883 


644809 


517781627 


28 3372.546 


9-2947671 


•001245330 


646416 


519718464 


28 3548938 


9-2986239 


-001243781 


648025 


521660125 


28-3725219 


93024775 


001242230 


(i49636 


523606()16 


28 3901391 


9-3063278 


001240095 


651249 


525557943 


284077454 


93101750 


0012391.57 


652864 


5275)4112 


284253408 


9-3140190 


-0012376-24 


654481 


529475129 


284429253 


9-3178599 


■00123G094 


650100 


531441000 


28-4694980 


9-3216975 


-001234308 


65772J 


533411731 


28-4780017 


9-3255320 


-001233U46 


G59344 


535387328 


28 49.50137 


9-3293634 


■001231527 


060969 


537307797 


285131.549 


9-3331916 


-001230012 


662596 


539353144 


285300852 


9-3370167 


-0012-28501 


064225 


541343375 


28-5482048 


9-3408336 


-001226994 


665850 


543338496 


28-50.57137 


9-.3446.575 


-001225490 


6o7489 


545338513 


28 5832119 


9-3484731 


001-2-23990 


009124 


547343432 


286000993 


9-352-2857 


-001222494 


070761 


549353259 


28-6181760 


9-3560952 


-001221001 


672400 


551368000 


28 6356421 


93599016 


-001219512 


674041 


553387661 


286530976 


9-3637049 


•001218027 


675684 


555412248 


286705424 


9-3675051 


-001216545 


677329 


557441767 


28 6879760 


9 3713022 


•001215007 


678976 


559476224 


28-7054002 


9 3750963 


-001213592 


680625 


501515625 


287228132 


9-3788873 


001-212121 


682276 


563559970 


287402157 


9-3826752 


001210654 


683929 


565609283 


28-7576077 


9-3864600 


-001209190 


685584 


567663552 


28-7749891 


9-3902419 


-001207729 


687241 


509722789 


28-7923601 


93940206 


-001206273 


688900 


571787000 


28-8097206 


9-3977964 


-001204819 


690561 


573850191 


28-8270706 


9-4015691 


-001203369 


692224 


575930308 


28-8444102 


9-4053387 


■001201923 


693889 


578009537 


28-8617394 


9-4091054 


-001200480 


695556 


580093704 


28-8790582 


9-4128690 


■001199041 


097225 


582182875 


28-8963666 


9-4100297 


■001197005 


098896 


584277050 


289136046 


9-4203873 


001190172 


700569 


586376253 


289309523 


9-4241420 


• 001194743 


702244 


588480472 


28 9482297 


9-4278936 


■001193317 


703921 


590589719 


28-9054907 


9-4316423 


■001191895 


705600 


592704000 


28 9827535 


9-4353880 


■001190476 


707231 


594823321 


290000000 


9-4391307 


■001189061 


708964 


596947083 


290172363 


9-4428704 


■001187648 


710049 


599077107 


29 0.344623 


9-4466072 


■001186240 


712336 


00121 1584 


29 0510781 


9-4.503410 


■001184834 


714025 


603351125 


290088837 


9-4540719 


■001183432 


715716 


005495736 


29 0860791 


9-4577999 


•001182033 


717409 


607045423 


291032644 


9-4615249 


•001180638 


719104 


009800192 


29-1204396 


9-4052470 


•001179-245 


7208fl 


011900049 


291370046 


9-4089661 


•001177856 


722500 


014125000 


291.547595 


9-4726824 


-001176471 


724201 


616295051 


291719043 


94763957 


•001175088 


725904 


618470208 


29 1890390 


9 4801061 


•001173709 


727609 


620650477 


29 2001037 


94838136 


•001172333 


729316 


622835864 


29-2232784 


9-4875182 


•001170960 


731025 


625026375 


29-2403830 


9 4912200 


-001169591 


732736 


627222016 


29-2574777 


9-4949188 


001168224 



176 



SQUARES, CUBES, ETC., OF NUMBERS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Koote. 


Reciprocals. 


"ssT 


734449 


629422793 


29-2745023 


9-4986147 


•001106801 


858 


736164 


631(128712 


29-2916370 


9-5023078 


•001165501 


859 


737881 


633839779 


29 .3087018 


9-5059980 


•001164144 


860 


739600 


636056000 


29-3257566 


9-.5096854 


•001162791 


861 


741321 


638277381 


29-3428015 


9-5133699 


•001161440 


862 


743044 


640503928 


29-3.598365 


9-5170515 


•001160093 


863 


744769 


642735647 


29-3768616 


9-5207303 


•001158749 


864 


746496 


644972.544 


29-3938769 


9.5244063 


•OOJ 1.57407 


865 


748225 


64721 4t)25 


29-5 108823 


9-52.80794 


•001 156069 


866 


749956 


64946189(1 


29-4278779 


9-5317497 


•001154734 


867 


751689 


6517143(53 


2944486.37 


9-5354172 


-001153403 


868 


753424 


653972032 


29-4618397 


9-5390818 


•001152074 


869 


755161 


656234909 


29-4788059 


9-5427437 


-001150748 


870 


756900 


658503000 


29-4957624 


9-5464027 


•001149425 


871 


758641 


660776311 


29-5127091 


9.5.500.589 


•001148106 


872 


760384 


663054848 


29-5296461 


9 5537123 


•001146789 


873 


762129 


665338617 


29-54(55734 


9-5573630 


•001145475 


874 


763876 


667627024 


29-5634910 


9-5610108 


•001144165 


875 


765625 


669921875 


29-5803989 


9-5646559 


•001142857 


876 


767376 


672221376 


29-5972972 


9-5682782 


•OOJ 14 1553 


877 


769129 


67452GJ33 


29-6141858 


9.5719377 


•001140251 


878 


770884 


676836152 


29-6310648 


9 5755745 


•001138952 


879 


77264J 


679151439 


296479.342 


9-5792085 


•0011.37656 


880 


774400 


681472000 


29 6647939 


9-5828397 


•001136364 


881 


7761(i] 


683797841 


29 6816442 


95864682 


-001135074 


882 


777924 


686128968 


29-6984848 


9-5900939 


•091133787 


883 


779689 


688465387 


29-7153159 


9-5937169 


•001132503 


884 


781456 


690807104 


29-7.321375 


9-5973373 


•0011.31222 


885 


783225 


693154125 


29-7489496 


9-6009548 


•001129944 


886 


7849'J6 


695506456 


29-7657521 


9-6045696 


-001128668 


887 


786769 


697864103 


29-7825452 


9-6081817 


001127396 


888 


788544 


700227072 


29-7993289 


9 6117911 


-001126126 


889 


790321 


7025953(i9 


29-8161030 


9-6153977 


•001124859 


890 


792100 


704969000 


29-8328678 


9-6190017 


•001123596 


891 


793881 


707347971 


29-8496231 


9-6226030 


-001122334 


892 


7956G4 


709732288 


29-8663690 


9-6262016 


•001121076 


893 


797449 


712121957 


29-8831056 


9 6297975 


•001119821 


894 


799236 


714516984 


29-8998328 


9-6333907 


•001118568 


895 


801025 


716917375 


29-9165506 


9-6369812 


•001117818 


896 


802816 


719323136 


29-9332591 


9-6405690 


-001116071 


897 


804609 


721734273 


29-9499583 


90441542 


•001114827 


898 


806404 


724150792 


29-9666481 


9-6477367 


•001113586 


899 


808201 


726572699 


29-9833287 


9-6513166 


•001112347 


900 


810000 


729000000 


30-0000000 


9-6548938 


•OOllllUl 


901 


811801 


731432701 


30-0166621 


9-6584684 


•OOJ 109878 


902 


813604 


733870808 


30-0333148 


9-6620403 


•001108647 


903 


8154U9 


736314327 


300499584 


9-6656096 


•001107420 


904 


817216 


738763264 


30-0665928 


9-6691762 


•001106195 


905 


819025 


741217625 


30-0832179 


9-6727403 


•001104972 


906 


820836 


743677416 


30-0998339 


9-6763017 


•001103753 


907 


822649 


746142643 


301164407 


9-6798604 


•001102.536 


908 


824464 


748613312 


30-1330383 


9-6834166 


001101322 


909 


826281 


751089429 


301496269 


9-686(^701 


•001 1001 10 


910 


828100 


753.571000 


301662063 


9-690.5211 


•00109890C 


911 


829921 


756058031 


30-1827765 


9-6940694 


•001097695 


912 


831744 


758550528 


301993377 


9-6976151 


•001096491 


913 


833569 


761048497 


30-2158899 


9-7011583 


•00109.5290 


914 


835396 


763.551944 


30-2324329 


9-7046989 


•OOJ 094092 


915 


837225 


76()060875 


30-2489669 


9-7082369 


00 J 092896 


916 


839056 


768575296 


30-2654919 


9-7117723 


-001091703 


917 


840889 


771095213 


30-2820079 


9-7153051 


-00 10! 105 13 


9J8 


842724 


773620632 


30-2985148 


9-7188354 


•001089325 



SQUARES, CUBES, ETC., OF NUMBERS. 



m 



No. 


Squares, 


Cubes. 


Square Roots. 


Oube Hoots, Reciprocals. 


919 


844561 


776151559 


30-3150128 


9-7223631 


001088139 


9-JO 


846400 


778688000 


30-3315018 


9-7258883 


001086957 


921 


848241 


781229961 


30-3479818 


9-7294109 


00108.5776 


922 


850084 


783777448 


30-3644529 


9-7329309 


001084599, 


923 


851929 


786330467 


30-3809151 


9-7364484 


001083423 


924 


853776 


788889024 


30-3973683 


9-7399634 


001082-251 


925 


855625 


791453125 


30-4J3,sj-27 


9-7434758 


001081081 


92G 


857476 


79402277G 


30-430-2481 


9-7469857 


001079914 


927 


859329 


796597983 


30-4466747 


9-7504930 


001078749 


928 


861184 


799178752 


30-4630924 


9-7539979 


001077586 


929 


863041 


801765089 


30-4795013 


9-7575002 


001070426 


930 


864900 


804357000 


30-4959014 


97610001 


001075-209 


931 


866761 


806954491 


30-5122926 


9-7644974 


001074114 


932 


868624 


809557568 


30-5286750 


9-7679922 


00 107-2901 


933 


870489 


812166237 


30-5450487 


9-7714845 


001071811 


934 


872356 


814780504 


30-5614136 


9-7749743 


001070064 


935 


874225 


817400375 


30-5777697 


9-7784616 


001069519 


936 


876096 


820025856 


30-5941171 


9-7819466 


001068376 


937 


877969 


822656953 


30-6104557 


9-7854288 


001067236 


938 


879844 


825293672 


30-6267857 


9-7889087 


001066098 


939 


881721 


827936J19 


30-6431069 


9-7923861 


001064963 


940 


883600 


830584tJcJ0 


30-6594 194 


9-7958611 


00 106383 J 


941 


885481 


833237621 


30-6757233 


9-7993336 


001002G99 


942 


887364 


835896888 


30-6920185 


9-8028036 


001001571 


943 


889249 


838561807 


30-7083051 


9-8002711 


00]G0o445 


944 


891136 


841232384 


30-7245830 


9-8097302 


001659322 


945 


893025 


843908625 


30-7408523 


9-8131989 


001(i.)8-2Jl 


940 


894916 


846590536 


30-7571130 


9-81065J1 


001057082 


947 


896869 


849278123 


30-7733651 


9-8-201109 


001055966 


948 


89i^7u4 


851971392 


30-7896086 


9-8-235723 


001054852 


949 


900601 


854670349 


30-8058436 


98270252 


001053741 


950 


902500 


857375000 


30-8220700 


9-8304757 


00105-2032 


951 


904401 


860085351 


30-8382879 


9-8339238 


001051525 


952 


906304 


862801408 


30-8544972 


9-8373695 


00105042J 


953 


908209 


865523177 


30-8706981 


9-8408127 


001(/4>JJl8 


954 


910116 


868250664 


30-8868904 


9-8442536 


001('48218 


955 


9l2u25 


870983875 


30-9030743 


9-8470920 


001047 120 


956 


913936 


873722816 


30-9192497 


9-8511280 


001046O-25 


957 


915849 


876467493 


3^-9354166 


9-8545617 


001044Ly.>2 


958 


917764 


879217912 


30-9515751 


9-857'Ji)29 


001043841 


959 


91'j681 


881974079 


30-9677-251 


9-8014218 


001042753 


96U 


921600 


884736000 


30-9838068 


9-8648483 


001041007 


961 


923521 


887503681 


31-0000000 


9-8682724 


001040583 


962 


925444 


890277128 


310101248 


9-8716941 


001039501 


963 


927369 


893056347 


31-03-2-2413 


9-8751135 


001038422 


964 


929296 


895841344 


31-0483494 


9-8785305 


001037344 


965 


931225 


898632125 


310644491 


9-8819451 


001030-209 


966 


933156 


901428696 


31-0805405 


9-8853574 


001035197 


967 


935089 


9042310G3 


310966236 


9-8887673 


001034120 


968 


937024 


907039232 


3 1 1126984 


9-8921749 


001033053 


969 


938961 


909853209 


3r 1287648 


9-8955801 


001031992 


970 


940900 


912673000 


311448230 


9-8989830 


001030928 


971 


942841 


915498611 


3l 1608729 


9-9023835 


001029866 


972 


944784 


918330048 


31 1769145 


9-9057817 


001028807 


973 


946729 


921167317 


3] 1929479 


9-9091776 


001027749 


974 


948676 


924010424 


31-2089731 


9-91257)2 


001026694 


975 


950625 


926859375 


31 2249900 


9-91596-24 


001025641 


976 


952576 


929714176 


31 2409987 


9-9193513 


001024590 


977 


954529 


932574833 


31-2569992 


9-9227379 


0010-23541 


978 


9564H4 


935441352 


312729915 


9-9261222 


001022495 


979 


958441 


938313739 


31-2889757 


9-9-295042 


001021450 


980 


960400 


941192000 


31 3049517 


9-9328839 


001020408 



178 



SQUARES, CUBES, ETC., OF NUMBERS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocala. 


981 


962361 


944076141 


31-3209195 


9-9362613 


001019168 


982 


964324 


946966168 


31-3338792 


9-9396.363 


•00101833ft 


983 


966289 


949862087 


31-3.528308 


9-!M3(t(l!l2 


-0111017294 


«84 


968256 


9527t)3!(l)4 


31-3687743 


9-9463797 


•001016260 


985 


970225 


955671625 


31-3847097 


9-9497479 


•001015228 


986 


972196 


958585256 


31.4006369 


9-9.531138 


•001014199 


987 


974169 


961504803 


31-416.5561 


9-9564775 


•001013171 


988 


976144 


964430272 


31-4.324673 


9-9.598389 


•001012146 


989 


978121 


967361669 


31-4483704 


9-9631981 


•001011122 


990 


980100 


970299000 


31-46426.54 


9-966.5549 


•001010101 


991 


982081 


973242271 


31-4801525 


9-9699095 


•001009082 


992 


984064 


976191488 


31-4960315 


9-9732(U9 


•001008065 


993 


986049 


979146657 


31-5119025 


9-9766120 


•001007049 


994 


988036 


982107784 


31-.i2776.55 


9-9799.599 


•001006036 


995 


990025 


985074875 


31.54.36206 


9-9833055 


001005025 


996 


992016 


988047936 


31 -.5,594677 


9-9866488 


•001004016 


997 


994009 


991026973 


31 -.57.53068 


9-9899900 


•001003009 


998 


996004 


994011992 


31-.591]380 


9-99.33289 


•001002004 


999 


998001 


997002999 


31-6069613 


9-9966656 


•001001001 


1000 


1000000 


1000000000 


31-6227766 


100000000 


•001000000 


1001 


1000201 


1003003001 


31-638.5840 


10-0033222 


•0009990010 


1002 


1004004 


1006012008 


31-6.543836 


10-0066822 


•0009980040 


1003 


1006009 


1009027027 


31.67017.52 


10-0099899 


•0009970090 


1004 


1008016 


1012048064 


31-6859.590 


10-01.331.55 


•0009960159 


1005 


1010025 


1015075125 


31-7017349 


100166.389 


•0009950249 


1006 


1012036 


1018108216 


31-717.5030 


100199601 


•0009940358 


1007 


1014049 


1021147343 


31-73.32633 


100232791 


•0009930487 


1008 


t016064 


1024192512 


31-74901.57 


10-02t>5958 


•0009920635 


1009 


1018081 


1027243729 


31-7647603 


10-0299104 


•0009910803 


1010 


1020100 


1030301000 


31-7804972 


10-0332228 


-0009900990 


1011 


1022121 


1033364331 


31-7962262 


100365330 


-0009891197 


1012 


1024144 


1036433728 


31-8119474 


10- 03984 10 


•0009881423 


1013 


1026169 


1039509197 


31-8276609 


10-0431469 


•0009871668 


10J4 


1028196 


1042590744 


31-8433686 


100464,506 


•0009861933 


10J5 


1030225 


1045678375 


318.590646 


100497.521 


•0009852217 


lOlG 


1032256 


1048772096 


31-8747.549 


10-0530514 


•0009842.520 


1017 


1034289 


1^51871913 


31-8904374 


100563485 


0009832842 


1018 


1036324 


1054977832 


31-9061123 


10-0.596435 


0009823183 


1019 


1038361 


10580898.59 


319217794 


10-0629364 


0009813543 


1020 


1040400 


1061208000 


31-9374.388 


l0-066i!271 


0009803922 


1021 


1042441 


1064332261 


31-95.30906 


1006951.56 


0009794319 


1022 


1044484 


1067462648 


31-9687347 


10-0728020 


0009784736 


1023 


1046529 


1070599167 


31-9843712 


10-07608()3 


0009775171 


A024 


1048576 


1073741824 


32-0000000 


100793684 


0009765625 


1025 


1050625 


1076890625 


320156212 


10-0826484 


0009756098 


1026 


1052676 


1080045576 


320312348 


10-0859262 


0009746589 


1027 


1054729 


1083206683 


320468407 


10-0892019 


0009737098 


1028 


1056784 


1086373952 


32-0624391 


10-0924755 


0009727626 


1029 


1058841 


1089547389 


32-0780298 


10-09.57469 


-0009718173 


1030 


10(50900 


1092727000 


32 0936131 


10-0990163 


•0009708738 


1031 


1062961 


1095912791 


32-1091887 


101022835 


•0009699321 


1032 


1065024 


1099104768 


32 1247568 


10-1055487 


-0009689922 


1033 


1067089 


1 102302937 


321403173 


10-1088117 


-0009680542 


1034 


1069156 


1105507304 


3215.58704 


101120726 


-0009671180 


1035 


1071225 


1108717875 


32-17141.59 


10-11.53314 


-0009661836 


1036 


1073296 


1111934656 


32- 1869539 


10-1185882 


1)009652510 


1037 


1075369 


11151.57653 


32-2024844 


10-1218428 


0009643202 


1038 


1077444 


1118386872 


32-2180074 


1012,50953 


0009633911 


1039 


1079521 


1121622319 


32 2335229 


10-1283457 


-0009624639 


1040 


1081600 


1124864000 


32-2490310 


10-1315941 


-0009615385 


1041 


1083681 


1128111921 


32-2645316 


10-1348403 


-0009606148 


1042 


1085764 


1131366088 


32-2800248 


10-1380845 


•0009596929 



TABLE XII. 

LOGAEITHMS OF NUMBERS 
FROM 1 TO 10000. 



TABLE, 



CONTAnrUTB 



THE LOGAEITHMS OE NUMBERS 



FROM 1 TO 10,000. 



NUMBERS FROM 1 TO 100 AND THEIR LOGARITHMS, 
WITH THEIR INDICES. 



Ka 


Logarithm. 


No. 


Logarithm. 


No. 


Logarithm. 


No. 


Logarithm. 


No. 


Logarithm. 


1 


0-ootiuuo 


21 


1-322219 


41 


1-612784 


61 


1-785330 


81 


1-908495 


2 


0-30 lUoO 


22 


1-342423 


42 


1-623249 


02 


1-792392 


82 


1-913814 


3 


0-477r21 


23 


1-301728 


43 


1-633468 


63 


1-799.341 


83 


1-919078 


4 


0-G020G0 


24 


1-380211 


44 


1-643453 


64 


1-806180 


84 


1-924279 


5 


0-698;)70 


25 


1-397940 


45 


1-653213 


65 


1-812913 


85 


1-D29419 


6 


0-778151 


20 


1-414973 


46 


1 •602758 


66 


1819.544 


86 


1-934498 


7 


0-845098 


27 


I-4313G4 


47 


1-672098 


67 


1-826075 


87 


1-039519 


8 


0-9030Ua 


28 


1-447158 


48 


1-681241 


68 


1-832509 


88 


1-944483 


9 


0-954243 


29 


1-402398 


49 


1-690190 


69 


1-838849 


89 


1-949390 


10 


1-0001)00 


30 


1-477121 


50 


1-098970 


70 


1-845098 


90 


1-954243 


11 


1-041393 


31 


1-491302 


51 


1-707570 


71 


1-851258 


91 


1-959041 


12 


1-079J81 


32 


1-505150 


52 


1-716003 


72 


1-857332 


92 


1-963788 


13 


1- 113943 


33 


1-518514 


53 


1-724270 


73 


1-8G3323 


93 


1-968483 


14 


1 14G128 


34 


1-53 J 479 


54 


1-732394 


74 


1-669232 


94 


1-973128 


15 


1-170091 


35 


1-544008 


55 


r 740303 


75 


1.875001 


95 


1-977724 


Hi 


1-204120 


3(5 


1-55G303 


50 


1-748188 


76 


1-880814 


96 


1-982271 


17 


1-23U449 


37 


1-568202 


57 


l-7.-)5875 


77 


1-88G491 


97 


1-986772 


18 


1-255273 


38 


1-579784 


58 


1-763428 


78 


1-892095 


98 


1-991226 


19 


1-278754 


39 


1-591065 


59 


1-770852 


79 


1-897627 


99 


1-995G35 


20 


1-301030 


40 


1-602060 


60 


1-778151 


80 


1-903090 


100 


2-000000 



Note. — In the following pare of the Table, the characterisrics are 
omitted, as they can be very easily supplied. Thus, the chaiacteristic 
of the logarithm of every integer number, consisting only of one number, 
isO; of two figures, 1; of three figures, 2; of four figui-es, 3; being always 
a uuit less than the number of flgiu-es contained in the integei- number. 



180 



LOGARITHMS OF NUMBERS. 



181 



Na 





1 


3 


3 


4 1 


5 


6 1 


7 


8 


9 (Diff 


l()0!(iOOOOO 


0004341000868 


001301 


001734 002166 002598 


003029 


003461 


003891 432 


1 


4321 


4751 


5181 


5609 


6038 6466 6894 


7321 


7748 


8174 42-1 


2 


8(i00 


9026 


9451 


9876 


01030U 010724 011147 


011570 


011993 


012415 424 


3 


012837 


013259 


013680 


014100 


452] 4940! 5360 


5779 


6197 


(5()I6!420 


4 


7033 


7451 


7868 


8284 


8700 9116 9532 


9947 


02U3(il 


020775 416 


5 


021189 


021603 


022016 


022428 


022841 023252 («36ti4 


024075 


448(5 


489li;412 


() 


5306 


5715 


6125 


6533 


6942 7350 7757 


HI (54 


8571 


8978; 408 


7 


9384 


9789 


030195 


030600 


031004 031408 031812 


03221(5 


032619 


033021,404 


8 033424 


033826 


4227 


4(i28 


5029 


5430 5830 


62:50 


(56211 


702H!40(» 


9 


7426 


7825 


8223 


8620 


9017 


9414' 9811 


04(J207 


040602 


040998:397 


110 


041393 


041787 


042182 


042576 


042969 043362 043755 


044148 


044540 


044932 393 


1 


53231 5714 


6105 


6495 


68851 7275! 7661 


8053 


8442 


H83:i MW 


2 


9218 9606 


9993 


050380 


050766 051153 051538 


051924 


0523(i'hf 


052(594 3Hii 


3 053078 10534(53 


053846 


4230 


4613 


4996 5378 


57(5(1, > 42 


()524 3s:{ 


4| 6905 7286 


7666 


8046 


8426 


8805 9185 


9563 


9942 


06032(1 379 


510600981061075 


061452 


061829 


062206 062582 062y.l8 


063;j33 


063709 


4083 37(; 


U; 4458 4832 


5206 


5580 


5953 i 6326! 6(599 


7071 


7443 


7815 373 


7 818!)! 8557 


8928 


9298 


9668 070038 070407 


07077(5 


071145 


071514 370 


8 071882 072250 


072617 


072985 


073352 3718 


4085 


4451 


4816 


5182 3ii(i 


9 


5547 5912 


6276 


6640 


7004 7368 


7731 


8094 


8457 


8819 3(53 


120 


07918J 079543!079904 


0802(56 


080626 080987 


081547 


081707 


082067 


082426 360 


1 


082785! 083 144 '083503 


3861 


4219, 4576 


4934 


5291 


5647 


6004 357 


2 


6360 67161 7071 


7426 


.7781 8136 


8490 


8845 


9198 


9552 355 


3 


9905 (190258 090611 


090963 


091315 091667 


092018 


092370 


092721 


093071 352 


4 


093422 37721 4122 


4471 


4820 5169 


5518 


58(56 


6215 


65(52 349 


5 


0910 


7257 


7604 


7951 


8298 8644 


8990 


9335 


9681 


1000261346 


6 


100371 


100715 


101059 


101403 


101747 102091 


102434 


102777 


103119 


3462! 343 


7 


3804 


4146 


4487 


4828 


51(59 5510 


5851 


6191 


6531 


6871 I341 


8 


7210 


7549 


7888 


»>27 


8565 8903 


9241 


9579 


9916 


1102531338 


9 


110590 


110926 


111263 


111599 


111934 


112270 


112605 


112940 


113275 


3609,335 


130 


] 13943 


114277 


114611 


114944 


11 5278 


115611 


115943 


11(5276 


116608 


116940 333 


1 


7271 


7603 


7934 


8265 


8595 


8926 


9256 


9586 


9915 


120245 1 330 


2 


120574 


120903 


121231 


121560 


121888 


122216 


122544 


122871 


123198 


35251328 


3 


3852 


4178 


4504 


4830 


5156 


5481 


5806 


6131 


6456 


67811325 


4 


7105 


7429 


7753 


8076 


8399 


8722 


9045 


9368 


9690 


1300121323 


5 


130334 


130655 


130977 


131298 


131619 


131939 


132260 


132580 


132900 


32191321 


6 


3539 


3858 


4177 


4496 


4814 


5133 


5451 


5769 


6086 


64031318 


7 


6721 


7037 


7354 


7671 


7987 


8303 


8618 


8934 


9249 


95641316 


8 


9879 1140194 


140508 


140822 


141136 


141450 


141763 


142076 


142389 


142702; 314 


9 


143015 3327 


3639 


3951 


4263 


4574 


4885 


5196 


5507 


5818311 


140 


146128 146438 


146748 


J 47058 


147367 


147676 


147985 


148294 


148603 


148911 309 


1 


9219 9527 


9835 


150142 


150449 


150756 


151063 


151370 


15167(5 


1519821307 


2 


152288 152594 


152900 


3205 


3510 


3815 


4120 


4424 


4728 


5032 305 


3 


5336 5640 


5943 


6246 


6549 


6852 


7154 


7457 


7759 


80611303 


4! 83621 8664 


ftj65 


9266 


9567 


9868 


1601(i8 


1604(59 


160769 


161068 |301 


5| 161368! 161667 


161967 


16226(5 


162:564 


162863 


31(51 


3460 


3758 


4055 1 299 


6 


4353 4650 


4947 


5244 


5541 


5838 


6134 


6430 


6726 


7022 '297 


7 


7317 7613 


7908 


8203 


8497 


8792 


9086 


9380 


9674 


99681295 


8 


170262 170555 


170848 


17114! 


171434 


171726 


172019 


172311 


172(503 


172895 293 


9 


3186 


^78 


3769 


4060 


4351 


4641 


4932 


5222 


5512 


5802 291 


150 ! 176091 


176381 


176670 


176959 


177248 


177536 


177825 


178113 


178401 


1786891289 


11 8077 


9264 


9552 


9839 


180126 


180413 


180699 


180986 


181272 


181558 287 


2 181844 


182129 


182415 


182700 


2985 


3270 


3555 


3839 


4123 


4407 285 


3i 4691 


49751 5259 


5542 


5825 


6108 


6391 


6674 


6956 


7239 283 


4| 752 J 1 78031 8084 


836(5 


8647 


8928 


9209 


9490 


9771 


190051 281 


5 190332 1906121 190892 


191171 


191451 


191730 


192010 


192289 


192567 


2846 279 


6 3125 3403! 3681 


3959 


4237 


4514 


4792 


5069 


5346 


5623 278 


7| 59UU' 6176! 6453 


6729 


7005 


7281 


7556 


7832 


8107 


8382 276 


8 86571 8932| 9206 


9481 


9755 


200029 


200303 


200577 


200850 


201124 274 


920139712016701201943 


202216 


202488 


2761 


3033 


3305 


3577 


3848,272 



No. I I 1 I 3 I 



I 4 I 5 I 



I 7 I 8 i 9 lOiff 



182 LOGARITHMS OF NUMBERS. 




Nal 1 1 1 a I 3 1 4: 1 5 1 6 1 7 1 8 1 9 lD.fl: 




160 204120 |204391| 


204663 


204934 


205204 


205475 


205746 


206016 


206286 


206556 271 




1 


6826 


7096 


7365 


7634 


7904 


8173 


8441 


8710 


8979 


9247 


269 




2 


9515 


9783 


210051 


210310 


210586 


210853 


211121 


211388 


211654 


211921 


267 




3 


21218;:^ 


212454 


2720 


2986 


3252 


3518 


3783 


4049 


4314 


4579 


266 




4 


4844 


5109 


5373 


5638 


5902 


6166 


6430 


6694 


6957 


7221 


264 




5 


7484 


7747 


8010 


8273 


8530 


8798 


9060 


9323 


9585 


9846 


262 




6 


220108 


220370 


220631 


220892 


221153 


221414 


221675 


221936 


222196 


222456 


261 




7 


2716 


2976 


3336 


3490 


3755 


4015 


4274 


4533 


4792 


5051 


259 




8 


5309 


5568 


5826 


6084 


6342 


6600 


6858 


7115 


7372 


7630 


258 




9 


7887 


8144 


8400 


8657 


8913 


9170 


9426 


9682 


9938 


230193 


'256 




170 


230449 


230704 


230960 


231215 


231470 


231724 


231979 


232234 


232488 


232742 


255 




1 


2996 


3250 


3504 


3757 


4011 


4264 


4517 


4770 


5023 


5276 


253 




2 


5528 


5781 


6033 


6285 


6537 


6789 


7041 


7292 


7544 


7795 


252 




3 


8046 


8297 


8548 


8799 


9049 


9299 


9550 


9800 


240050 


240300 


250 




4 


240549 


240799 


241048 


24 J 297 


241546 


241795 


242044 


242293 


2541 


2790 


249 




5 


3038 


3286 


3534 


3782 


4030 


4277 


4525 


4772 


5019 


5266 


248 




6 


5513 


5759 


6006 


6252 


6499 


6745 


6991 


7237 


7482 


7728 


246 




7 


7973 


8219 


8464 


8709 


8954 


9198 


9443 


9687 


9932 


250176 


245 




8 


250420 


250664 


250908 


251151 


251395 


251638 


251881 


252125 


252368 


2610 


243 




9 


2853 


3096 


3338 


3580 


3822 


4064 


4300 


4548 


4790 


5031 


242 




180 


255273 


255514 


255755 


255996 


256237 


256477 


256718 


256958 


257198 


257439 


241 




1 


7679 


7918 


8158 


8398 


8637 


8877 


9116 


9355 


9594 


9833 


239 




2 


260071 


260310 


260548 


260787 


261025 


261263 


261501 


261739 


261976 


262214 


238 




3 


2451 


2688 


2925 


3162 


3399 


3636 


3873 


4109 


4346 


4582 


237 




4 


4818 


5054 


5290 


5525 


5761 


5996 


6232 


6467 


6702 


6937 


235 




5 


7172 


7406 


7641 


7875 


8110 


8344 


8578 


8812 


9046 


9279 


234 




6 


9513 


9746 


9980 


270213 


270446 


270679 


270912 


271144 


271377 


271609 


233 




7 


271842 


272074 


272306 


2538 


2770 


3001 


3233 


3464 


3696 


3927 


232 




8 


4158 


4389 


4620 


4850 


5081 


5311 


5542 


5772 


6002 


6232 


230 




9 


6462 


6692 


6921 


7151 


7380 


7609 


7838 


8067 


8296 


8525 


229 




190 


278754 


278982 


279211 


279439 


279667 


279895 


280123 


280351 


280578 


280806 


228 




1 


281033 


281261 


281488 


281715 


281942 


282169 


2396 


2622 


2849 


3075 


227 




2 


3301 


3527 


3753 


3979 


4205 


4431 


4656 


4882 


5107 


5332 


226 




3 


5557 


5782 


6007 


6-232 


6456 


6681 


6905 


7130 


7354 


7578 


225 




4 


7802 


8026 


8249 


8473 


8696 


8920 


9143 


9306 


9539 


9812 


223 




5 


290035 


29U257 


290480 


290702 


290923 


291147 


291369 


291591 


291813 


292034 


222 




6 


2256 


2478 


2699 


2920 


3141 


3363 


3584 


3804 


4025 


4246 


221 




7 


4466 


4687 


4907 


5127 


5347 


5567 


5787 


6007 


6226 


6446 


220 




8 


6665 


6884 


7104 


7323 


7542 


7761 


7979 


8198 


8416 


8635 


219 




9 


8853 


9071 


9289 


9507 


9725 


9943 


300161 


300378 


300595 


300813 


218 




m 


301030 


301247 


301464 


301681 


301898 


302114 


302331 


302547 


302764 


302980 


217 




1 


3196 


3412 


3628 


3844 


40&9 


4275 


4491 


4706 


4921 


5136 


216 




2 


5351 


5566 


5781 


5996 


6211 


6425 


6639 


6854 


7068 


■ 7282 


215 




3 


7496 


7710 


7924 


8137 


8351 


8564 


8778 


8991 


9204 


9417 


213 




4 


9630 


9843 


310056 


310268 


310481 


310693 


310906 


311118 


311330 


311542 


212 




5 


311754 


311966 


2177 


2389 


2600 


2812 


3023 


3234 


3445 


3656 


211 




6 


3867 


4078 


4289 


4499 


4710 


4920 


5130 


5340 


5551 


5760 


210 




7 


5970 


6180 


6390 


6599 


6809 


7018 


7227 


7436 


7646 


7854 


209 




8 


8063 


8272 


8481 


8689 


8898 


9106 


9314 


9522 


9730 


9938 


208 







320146 


320354 


320562 


320769 


320977 


321184 


321391 


321598 


321805 


322012 


207 




210 


322219 


322426 


322633 


322839 


323046 


323252 


323458 


323665 


323871 


324077 


206 




1 


4282 


4488 


46:)4 


4899 


5105 


5310 


5516 


5721 


5926 


6131 


205 




2 


6336 


6541 


6745 


6950 


7155 


7359 


7563 


7767 


7972 


8176 


204 




3 


8380 


8583 


8787 


8i*91 


9194 


9398 


9601 


9805 


330008 


330211 


203 




4 


330414 


330617 


330819 


331022 


331225 


331427 


331630 


331832 


2034 


2236 


202 




5 


2438 


2640 


2842 


3044 


3246 


3447 


3649 


3850 


4051 


4253 


202 




6 


4454 


4055 


4850 


5057 


5257 


5458 


5658 


5859 


6059 


6260 


201 




7 


6460 


6660 


6860 


7060 


7260 


7459 


7659 


7858 


8058 


8257 


200 




8 


8456 


8656 


8855 


9054 


9253 


9451 


9650 


9849 


340047 


340246 


199 




9 


340444 


340642 


340841 


341039 


341237 


341435 


341632 


341830 


2028 


2225 


198 




No. 


1 


1 1 


» a 


1 3 


J 4 


1 5 


1 6 


1 'i 


I 8 


1 9 


'Dili 

1 





LOGARITHMS OF NUMBERS. 



isa 



No. 1 


1 


1 1 


3 1 


3 1 


4 1 


5 • 


6 1 


7 1 


8 


9 


Ditt 


220 


342423 


342620 


342817 


343014, 


J43212 ;:43409 


343606 


•J43802 


343999 


344196 


i97 


1 


4392 


4589 


4785 


4981 


51781 


5374 


5570 


5766 


5962 


6157 


196 


o 


6353 


6549 


6744 


6939 ! 


7i:r)i 


7330 


7525 


7720 


7915 


8110 


195 


3 


8305 


8500 


8694 


8889 


9083 


9278 


9472 


966(5 


9860 


350054 


194 


4 


350248 


350442 


350636 


350829 


551023 


351216 


351410 


351603 


351796 


1989 


193 


5 


2183 


2375 


2568 


2761 


2954 


3147 


3339 


3532 


37-24 


3916 


193 


6 


4108 


4301 


4493 


4685 


4876 


5068 


5260 


5452 


5643 


5834 


192 


7 


6026 


6217 


6408 


6599 


6790 


1)981 


7172 


7363 


7554 


7744 


191 


8 


7935 


8125 


8316 


8506 


8696 


i'.Am 


9076 


926() 


9456 


9646 


190 


9 


9835 


360025 


360215 


360404 


360593 


360783 


360972 


361161 


361350 


361539 


189 


230 


361728 


361917 


362105 


362294 


362482 


362671 


362859 


363048 


36;»e36 


363424 


188 


1 


3612 


38UI) 


3988 


4176 


4363 


4551 


4739 


4926 


5 J 13 


5301 


188 


2 


5488 


5675 


5802 


6049 


6236 


6423 


6610 


6706 


6983 


7169 


187 


3 


7356 


7542 


7729 


7915 


8101 


82^7 


8473 


8659 


8845 


9030 


186 


4 


9216 


9401 


9587 


9772 


9958 


370143 


370328 


370513 


376()98 


370883 


185 


5 


371068 


371253 


371437 


371622 


371806 


1991 


2175 


2360 


2544 


2728 


184 


G 


21)12 


30^6 


3280 


341)4 


3647 


3831 


4-015 


4198 


4382 


4565 


184 


7 


4748 


4932 


5115 


5298 


5481 


5664 


5846 


6029 


6212 


6394 


183 


8 


6577 


6759 


6942 


7124 


7306 


7488 


7670 


7852 


8634 


8216 


182 


9 


8398 


8580 


8761 


8943 


9124 


9306 


9487 


9668 


9849 


380030 


181 


24;) 


380211 


380392 


380573 


380754 


380934 


381115 


381296 


381476 


381656 


381837 


181 


1 


2017 


2197 


2377 


2557 


2737 


2917 


:i097 


3277 


3456 


3636 


18C 


2 


3815 


3995 


4174 


4353 


4533 


4712 


4891 


5070 


5249 


5428 


179 


3 


5606 


5785 


5964 


6142 


6321 


6499 


6677 


6856 


7034 


7212 


178 


4 


7390 


7568 


7746 


71.^^3 


8101 


8279 


8456 


8634 


8811 


8989 


178 


5 


9166 


9343 


9520 


96981 


9875 


390051 


390228 


390405 


390582 


390759 


177 


6 


390935 


391112 


391288 


391464 


391641 


1817 


1993 


2169 


2345 


2521 


176 


7 


2()97 


2873 


3048 


3224 


3400 


3575 


3751 


3926 


411)1 


4277 


176 


8 


4452 


4627 


4802 


4977 


5152 


5320 


5501 


5676 


5850 


6025 


175 


9 


6199 


6374 


6548 


6722 


6896 


7071 


7245 


7419 


7592 


77(56 


174 


250 


397940 


398114 


398287 


398461 


398634 


398808 


398981 


399154 


399328 


399501 


173 


1 


9674 


9847 


400020 


400192 


400365 


400538 


400711 


400883 


401056 


401228 


173 


2 


401401 


401573 


1745 


1917 


2089 


2261 


2433 


2605 


2777 


2949 


172 


3 


3121 


3292 


3464 


3635 


3807 


3978 


4149 


4320 


4492 


4663 


171 


4 


4834 


5005 


5176 


5346 


5517 


5688 


5858 


6029 


6199 


6370 


171 


5 


6540 


6710 


6881 


7051 


7221 


7391 


■7561 


7731 


7901 


8070 


170 


6 


8240 


8410 


8579 


8749 


8918 


9087 


9257 


9426 


9595 


9764 


169 




9933 


410102 


410271 


410440 


410609 


410777 


410046 


411114 


4.11283 


411451 


169 


8 


411620 


1788 


19.--6 


2124 


2293 


2461 


2629 


2706 


2.-)l4 


3132 


168 


9 


3300 


3467 


3635 


3803 


3970 


4137 


43',(5 


4472 


463!) 


4806 


167 


26,) 


414973 


415140 


415307 


415474 


415641 


415808 


415974 


416141 


4163()f^ 


41(5474 


167 


J 


6641 


6807 


6973 


7139 


7306 


7472 


7t)38 


7804 


:!»70 


8135 


166 


2 


8301 


8467 


8633 


8798 


8964 


9129 


9295 


946u 


'.)(i25 


9791 


165 


3 


9956 


420121 


4202S6 


420451 


420616 


420781 


420945 


421110 


421275 


421439 


165 


4 


421604 


1768 


1933 


2097 


2261 


2426 


2590 


2754 


2918 


3082 


164 


5 


3246 


3410 


3574 


3737 


3901 


4065 


4228 


4392 


4555 


4718 


164 


6 


4882 


5045 


5208 


5371 


5534 


5697 


5860 


6023 


61W6 


))349 


163 


' 


6511 


6674 


6836 


6999 


7161 


7324 


7486 


7648 


7811 


7973 


162 


8 


8135 


8297 


8459 


8621 


8783 


8944 


9106 


9268 


942J 


9591 


162 


9 


9752 9914 


430075 


430236 


430398 


430559 


430720 


430881 


431042 


431203 


161 


270 


431364 431525 


431685 


431846 


432007 


432167 


432328 


4324HH;432li4!» 


432H09 


161 


1 


29691 3i:i0 


32J,) 


3450 


3610 


3770 


3930 


4iV.«); 4241) 4409 


160 


2 


45691 4729 


4888 


5048 


5207 


5367 


5526 


5685 


5844, 6004 


159 


3 


61631 6322 


6481 


6640 


6799 


6957 


7116 


7275 


7433 75921 159 


4 


7751 


7909 


8067 


8226 


8384 


8542 


8701 


8859 


9.17 


9175 


158 


5 


9333 


9491 


9648 


9806 


9964 


440122 


440279 


440437 


440594 


440752 


158 


6 


440909 


441066 


441224 


441381 


441538 


1695 


1652 


2009 


2166 


2323 


157 


7 


248^ 


2637 


2793 


2950 


3106 


3263 


3419 


3576 


3732 


3889 


157 


£ 


404i 


j 4201 


4357 


4513 


4669 


4825 


49^1 


5137 


5293 


5449 


156 


S 


5604 


1 5760 


5915 


6071 


6226 


6382 


6537 


6692 


6848 


7003 


155 



No. |011|31314:|5|6|7|8|9 



Die 



184 



LOGARITHMS OF NUMBERS. 



No. 





1 


3 


3 


^ 


5 


G 


7 


8 


9 


D,ff, 


280 447158 


447313 447468 


447623 


447778 


447933 


448088 


448242 


448397 


448552 


155 


1 870G 


8861 


9015 


9170 


93-24 


9478 


9033 


9787 


3341 


450095 


154 


2 4.:j:4D 


453403 


450557 


453711 


450805 


451010 


451172 


451320 


451479 


1033 


154 


3 


J78.'i 


1940 


2033 


2247 


2400 


2553 


2706 


2859 


3012 


3165 


153 


4 


3 J 18 


3471 


3o24 


3777 


3930 


4032 


4235 


4387 


4540 


4632 


l.':3 


5 


4845 


4997 


5150 


5302 


5454 


5;i0{; 


575(1 


5310 


6062 


6214 


152 


6 


onoa 


0518 


6070 


6821 


6373 


7125 


7276 


7428 


7579 


7731 


152 


7 


7c^82 


80r,3 


8104 


8336 


8487 


8038 


8783 


8940 


9091 


9-242 


151 


8 


9332 


9543 


on. 14 


9845 


9935 


460146 


46023() 


460447 


460597 


460748 


151 


9 


460898 


461048 


4GU38 


401348 


461499 


1649 


1793 


1948 


2098 


2248 


150 


290 402338 


462548 


402337 


402>S47 


462997 


463140 


463296 


4(53445 


403594 


463744 


150 


i; 38a3 


4042 


4131 


4340 


4430 


4633 


4788 


4936 


5085 


5234 


149 


2 5383 


5532 


5i)«0 


5.-3 


5377 


612.; 


6274 


6423 


6571 


6719 


149 


3 


6868 


7.116 


7164 


7312 


7400 


7608 


7756 


7904 


8052 


8200 


148 


4 


8347 


8405 


8043 


8790 


8938 


9085 


9233 


9380 


9527 


9675 


148 


5 


9822 


9963 


470116 


470263 


4704 Iv) 


470557 


470704 


470851 


470998 


471145 


147 


6 


471292 


471438 


1585 


1732 


1S78 


2025 


2171 


2318 


2464 


2010 


146 


7 


2756 


2903 


3043 


3195 


334] 


3487 


?633 


3779 


3925 


4071 


146 


8 


4216 


4362 


4508 


4J53 


4793 


4344 


5090 


5235 


5381 


5526 


146 


9 


5671 


5816 


5962 


6107 


6252 


6397 


6542 


6687 


6832 


6376 


145 


300 


477121 


477266 


477411 


477555 


477700 


477844 


477989 


478133 


478278 


478422 


145 


1 


8560 
48000f 


8711 


8855 


8999 


9143 


9287 


9431 


9575 


9719 


9863 


144 


2 


480151 


480294 


480438 


480582 


480725 


480869 


481012 


481156 


481299 


144 


3 


1443 


1586 


1729 


1872 


2016 


2159 


2302 


2445 


2588 


2731 


143 


4 


2874 


3016 


3159 


3302 


3445 


3587 


3730 


3872 


4015 


4157 


143 


5 


4300 


4442 


4585 


4727 


4869 


5011 


5153 


5295 


5437 


5579 


142 


6 


5721 


5863 


6005 


6147 


6289 


6430 


6572 


6714 


6855 


6997 


142 


7 


7138 


7280 


7421 


7563 


7704 


7845 


7986 


8127 


8269 


8410 


141 


8 


8551 


8692 


8833 


8974 


9114 


9255 


9396 


9537 


9677 


9813 


141 


9 


9958 


490099 


490239 


490380 


490520 


490661 


490801 


490941 


491081 


491222 


140 


310 


491362 


491502 


491642 


491782 


491922 


492062 


492201 


492341 


492481 


492621 


140 


1 


2760 


2900 


3040 


3179 


3319 


3458 


3597 


3737 


3876 


4015 


139 


o 


4155 


4294 


4433 


4572 


4711 


4850 


4989 


5128 


5267 


5406 


139 


3 


5544 


5683 


5822 


5960 


6099 


6238 


6376 


6515 


6653 


6791 


139 


4 


6930 


7068 


7206 


7344 


7483 


7621 


7759 


7897 


8035 


8173 


138 


5 


8311 


8448 


8586 


8724 


8862 


8999 


9137 


9275 


9412 


9550 


138 


6 


9687 


9824 


9962 


500099 


500236 


500374 


500511 


500648 


500785 


500922 


137 


7 


501059 


501196 


501333 


1470 


1()07 


1744 


1880 


2017 


2154 


2291 


137 


8 


2427 


2564 


2700 


2837 


2973 


3109 


3246 


3382 


3518 


3655 


136 


9 


3791 


3927 


4063 


4199 


4335 


4471 


4607 


4743 


4878 


5014 


136 


320 


505150 


50528() 


505421 


505557 


505693 


505828 


505964 


506099 


506234 


506370 


136 


1 


6505 


6640 


6776 


6911 


7046 


7181 


7316 


7451 


7586 


7721 


135 


2 


7856 


7991 


8126 


8260 


8395 


8530 


8664 


8799 


8934 


9068 


135 


3 


9203 


9337 


9471 


9606 


9740 


9874 


510009 


510143 


510277 


510411 


134 


4 


510545 


510679 


510813 


510947 


511081 


511215 


1349 


1482 


1616 


1750 


134 


5 


1883 


21)17 


2151 


2284 


2418 


2551 


2684 


2818 


2951 


3084 


133 


6 


3218 


3351 


3484 


3617 


3750 


3883 


4016 


4149 


4282 


4415 


133 


7 


4548 


4681 


4813 


4946 


5079 


5211 


5344 


5476 


5609 


5741 


133 


8 


5874 


60!)6 


6133 


6271 


6403 


6535 


6668 


6800 


6932 


7064 


132 


9 


7196 


7328 


7460 


7592 


7724 


7855 


7987 


8119 


8251 


8382 


132 


330 


518514 


518646 


518777 


518909 


519040 


519171 


519303 


519434 


519566 


519697 


131 


1 


9828 


9959 


520030 


520221 


520353 


520484 


520615 


520745 


520876 


521007 


131 


2 


521 KW 


521269 


1430 


1530 


1661 


1792 


1922 


2053 


2183 


2314 


13J 


3 


2444 


2575 


2705 


2835 


2966 


3090 


3226 


3356 


3486 


3616 


130 


4 


374(i 


3876 


40J6 


4136 


4266 


4396 


4526 


4656 


4785 


4915 


130 


5 


5045 


5174 


53()4 


5434 


5563 


5693 


5822 


5951 


6081 


6210 


129 


6 


6339 


6463 


653H 


6727 


6856 


6985 


7114 


7243 


7372 


7501 


129 


7 


7630 


7759 


7888 


8016 


8145 


8274 


8402 


8531 


8660 


8788 


129 


8 


8917 


9045 


9174 


9302 


9430 


9559 


9687 


9815 


9943 


530072 


128 


9 


530200 


530320 


530456 


530584 


530712 


530840 


530968 


531096 


531223 


1351 


128 



Wd.|0I119I3|4:|$ 



r I 8 I 9 iDifl 



LOGARITHMS OF NUMBERS. 



185 



3 I 



I 7 I 8 / 9 iDiffi 



340 


5J1-17Q 


531607 


531734 


5318621 531990 


532117 


532245 


532372 


532500 


532627 


128 


1 


2754 


2882 


3009 


313()! 3264 


3391 


3518 


3645 


3772 


3899 


127 


'J 


4026 


4153 


4280 


4407 


4534 


4661 


4787 


4914 


5041 


5167 


127 


3 


5294 


5421 


5547 


5674 


5800 


5927 


6053 


6180 


6306 


6432 


126 


4 


6558 


6685 


6811 


6937 


7063 


7189 


7315 


7441 


7567 


7693 


126 


5 


7819 


7945 


8071 


8197 


8322 


8448 


8574 


8699 


8825 


"951 


126 


6 


9076 


9202 


9327 


9452 


9578 


9703 


9829 


9954 


540079 


540204 


125 


7 


540329 


540455 


540580 


540705 


540830 


540955 


541080 


541205 


1330 


1454 


125 


8 


1579 


1704 


1829 


1953 


2078 


2203 


2327 


2452 


2576 


2701 


125 


9 


2825 


2950 


3074 


3199 


3323 


3447 


3571 


3696 


3820 


3944 


124 


350 


544068 


544192 


544316 


544140 


544564 


544688 


544812 


544936 


545060 


545183 


124 


1 


5307 


5431 


5555 


5678 


5802 


5925 


6049 


6172 


6296 


6419 


124 


2 


6543 


6666 


6789 


6913 


7036 


7159 


7282 


7405 


7529 


7652 


123 


3 


7775 


7898 


8021 


8144 


8267 


8389 


8512 


8635 


8758 


8881 


123 


4 


9003 


9126 


9249 


9371 


9494 


9616 


9739 


9861 


9984 


550106 


123 


5 


550228 


550351 


550473 


550595 


550717 


550840 


550962 


551084 


551206 


1328 


122 


6 


1450 


1572 


1694 


1816 


1938 


2060 


2181 


2303 


2425 


2547 


122 


7 


2668 


2790 


2911 


3033 


3155 


3276 


3398 


3519 


3640 


3762 


121 


8 


3883 


4004 


4126 


4247 


4368 


4489 


4610 


4731 


4852 


4973 


121 


9 


5094 


5215 


5336 


5457 


5578 


5699 


5820 


5940 


6061 


6182 


121 


360 


556303 


556423 


556544 


556664 


556785 


556905 


557026 


557146 


557267 


557387 


120 


1 


7507 


7627 


7748 


7868 


7988 


8108 


8228 


8349 


8469 


8589 


120 


2 


8709 


8829 


8948 


9068 


9188 


9308 


9428 


9548 


9667 


9787 


120 


3 


9907 


560026 


560146 


560265 


560385 


560504 


560624 


560743 


560863 


560982 


119 


4 


561101 


1221 


1340 


1459 


1578 


1698 


1817 


1936 


2055 


2174 


119 


5 


2293 


2412 


2531 


2650 


2769 


2887 


3006 


3125 


3244 


3362 


119 


6 


3481 


3600 


3718 


3837 


3955 


4074 


4192 


4311 


4429 


4548 


119 


7 


4666 


4784 


4903 


5021 


5139 


5257 


5376 


5494 


5612 


5730 


118 


8 


5848 


5966 


6084 


6202 


6320 


6437 


6555 


6673 


6791 


6909 


118 


9 


7026 


7144 


7262 


7379 


7497 


7614 


7732 


7849 


7967 


8084 


118 


370 


568202 


568319 


568436 


568554 


568671 


568788 


568905 


569023 


569140 


569257 


117 


1 


9374 


9491 


9608 


9725 


9842 


9959 


570076 


570193 


570309 


570426 


117 


2 


570543 


570660 


570776 


570893 


571010 


571126 


1243 


1359 


1476 


1592 


117 


3 


1709 


1825 


1942 


20.58 


2174 


2291 


2407 


2523 


2639 


2755 


116 


4 


2872 


2988 


3104 


3220 


3336 


3452 


3568 


3684 


3800 


3915 


116 


5 


4031 


4147 


4263 


4379 


4494 


4610 


4726 


4841 


4957 


5072 


116 


6 


51G8 


5303 


5419 


5534 


5650 


5765 


5880 


5996 


6111 


6226 


115 


7 


6341 


6457 


6572 


6687 


6802 


6917 


7032 


7147 


7262 


7377 


115 


8 


7492 


7607 


7722 


7836 


7951 


8066 


8181 


8295 


8410 


8525 


115 


9 


8639 


8754 


8868 


8983 


9097 


9212 


9326 


9441 


9555 


9669 


114 


380 


579784 


579898 


580012 


580126 


580241 


580355 


580469 


580583 


580697 


580811 


114 


1 


580925 


581039 


1153 


1267 


1381 


1495 


1608 


1722 


1836 


1950 


114 


2 


2063 


2177 


2291 


2404 


2518 


2631 


2745 


2858 


2972 


3085 


114 


3 


3199 


3312 


3426 


3539 


3652 


3765 


3879 


3992 


4105 


4218 


113 


4 


4331 


4444 


4557 


4670 


4783 


4896 


5009 


5122 


5235 


5348 


113 


5 


5461 


5574 


5686 


5799 


5912 


6024 


6137 


6250 


6362 


6475 


113 


6 


6587 


6700 


6812 


6925 


7037 


7141) 


7262 


7374 


7486 


7599 


112 




7711 


7823 


7935 


8047 


8160 


8272 


8384 


8496 


8608 


8720 


112 


8 


8832 


8944 


9050 


9167 


9279 


9391 


9503 


9615 


9726 


9838 


112 


9 


9950 


590061 


590173 


590284 


590396 


590507 


590619 


590730 


590842 


590953 


112 


390 


591065 


591176 


591287 


591399 


591510 


591621 


591732 


591843 


591955 


592066 


111 


1 


2177 


2288 


23a9 


2510 


2621 


2732 


2843 


2954 


3064 


3175 


111 




3286 


3397 


35U« 


3618 


3729 


3840 


3950 


4061 


4171 


4282 


111 


3 


4393 


4503 


4614 


4724 


4834 


4945 


5055 


5165 


5276 


5386 


110 


4 


5496 


5006 


5717 


5827 


5937 


6047 


6157 


6267 


6377 


6487 


110 


5 


6597 


6707 


6817 


6927 


7037 


7146 


7256 


7366 


7476 


7586 


110 


6 


7695 


7805 


7914 


8024 


8134 


8243 


8353 


8462 


8572 


8681 


110 


7 


8791 


89C0 


9009 


9119 


9228 


9337 


9446 


9556 


9665 


9774 


109 


8 


9883 


9992 


600101 


600210 


600319 


600428 


600537 


600646 


600755 


600864 


109 


9 


600973 


601082 


1191 


1299 


1408 


1517 


1625 


1734 


1843 


1951 


109 



W.|0|l|a|3|4:|5|6|7| 



I 9 ID* 



186 



LOGARITHMS OF NUMBERS. 



No. I I- 1 I 



I 3 I 4 I 



I 6 I 7 I 8 I 9 iDifS 



400 


602060 


602169 


602277 


602386 


602494 


502603 


602711 602819] 


602928 


603038 


108 


1 


3144 


3253 


3361 


3469 


3577 


3638 


3794 39021 


4010 


4118 


108 


2 


4226 


4334 


4442 


4550 


4658 


4766 


4874 


4982 


5089 


5197 


108 


n 


5305 


5413 


5521 


5r.28 


5736 


5844 


5951 


6059 


6166 


6274 


108 


4 


6381 


6489 


6596 


6704 


6811 


6919 


7026 


7133 


7241 


7348 


107 


5 


7455 


7562 


7669 


7777 


7884 


7991 


8098 


8205 


8312 


8419 


107 


6 


8526 


8633 


8740 


8847 


8954 


9061 


91G7 


9274 


9381 


9488 


107 


7 


9594 


9701 


9808 


9914 


610021 


610128 


610234 


610341 


6104-7 


G1053-; 


107 


8 


610660 


6107G7 


610873 


610970 


1086 


1192 


1298 


1405 


1511 


1617 


106 


9 


1723 


1829 


1933 


2042 


2148 


2254 


2360 


2466 


2572 


2678 


106 


410 


612784 


612890 


612996 


613102 


613207 


613313 


613419 


613525 


613630 


613736 


106 


1 


3842 


3947 


4053 


4159 


4264 


4370 


4475 


4581 


4688 


4792 


106 




4897 


5003 


5108 


5213 


5319 


5424 


5529 


5G34 


5740 


5843 


105 


3 


5950 


6055 


6160 


6265 


6370 


6476 


6581 


6686 


0790 


6895 


105 


4 


7000 


7105 


7210 


7315 


7429 


7525 


7829 


7734 


7839 


7943 


105 


5 


8048 


8153 


8257 


8362 


8466 


8571 


8676 


8780 


8884 


8989 


105 


6 


9093 


9198 


9302 


9406 


9511 


9615 


9719 


9824 


9928 


620032 


104 


7 


620136 


620240 


620344 


620448 


620552 


62065S 


620700 


623864 


0209G8 


1072 


104 


8 


117G 


1280 


1384 


1488 


1592 


1695 


1709 


1903 


2007 


2110 


104 


9 


2214 


2318 


2421 


2525 


2628 


2732 


2835 


2939 


3342 


314G 


104 


420 


023249 


623353 


62345G 


623559 


623663 


623766 


623869 


G23973 


624076 


624179 


103 


1 


4282 


4385 


4488 


4591 


4695 


4798 


4901 


5004 


5107 


5210 


103 


2 


5312 


5415 


5518 


5621 


5724 


5827 


5929 


6032 


6133 


6238 


103 


3 


6340 


6442 


6546 


6648 


6751 


6853 


6956 


7058 


716! 


7263 


103 


4 


7366 


7468 


7571 


7673 


7775 


7878 


7980 


8082 


8185 


8287 


102 


5 


8389 


8491 


8593 


8695 


8797 


8900 


9092 


9104 


920G 


9308 


102 


6 


9410 


9512 


9613 


9715 


9817 


9919 


630021 


G30123 


630224 


630323 


102 


7 


G30428 


630530 


630631 


630733 


630835 


630936 


1038 


1139 


1241 


1342 


102 


8 


^-14 


1545 


1647 


1748 


1849 


1951 


2352 


2133 


2235 


2336 


131 


9 


2437 


2559 


2660 


2761 


2862 


2963 


30. ;4 


3165 


3266 


33G7 


101 


430 


G33468 


633569 


633G70 


633771 


633872 


633973 


634074 


634175 


634276 


634376 


101 


1 


4477 


4578 


4679 


4779 


4880 


4981 


5081 


5182 


5283 


5383 


101 


2 


5484 


5584 


5685 


5785 


5886 


5986 


6087 


6187 


6287 


6388 


100 


3 


6488 


0588 


G688 


6789 


6889 


6989 


70S9 


7189 


7290 


7390 


100 


4 


7490 


7590 


7690 


7793 


7890 


7990 


8090 


8190 


8290 


83H9 


100 


5 


8489 


853J 


81389 


8789 


8838 


8988 


90,88 


9188 


9287 


9337 


100 





9486 


9386 


9686 


9785 


9885 


9984 


640084 


640183 


640283 


040382 


99 


7 


640481 


640581 


640680 


640779 


640879 


640978 


1077 


1177 


1276 


1375 


99 


8 


1474 


1573 


1672 


1771 


1871 


1970 


2069 


2168 


2267 


2306 


99 


9 


2465 


2563 


2662 


2761 


2860 


2959 


3058 


3156 


3255 


3354 


99 


440 


643453 


643551 


643650 


643749 


643847 


643946 


644044 


644143 


644242 


644340 


98 


1 


4433 


4537 


4636 


4734 


4832 


4931 


5029 


5127 


5226 


5324 


.98 




5422 


5521 


5619 


5717 


5815 


5913 


6011 


6110 


6208 


6306 


98 


3 


6404 


C5!J2 


6600 


6698 


6796 


6894 


6992 


7089 


7187 


7285 


98 


4 


7383 


7481 


7579 


7676 


7774 


7872 


7969 


8067 


8165 


8362 


98 


5 


8360 


8458 


8555 


8653 


8750 


8848 


8945 


9043 


9140 


9237 


97 


6 


9335 


9432 


9530 


9627 


9724 


9821 


9919 


650016 


650113 


650210 


97 


7 


650308 


650405 


650502 


650599 


650696 


G50793 


650890 


0987 


1084 


1181 


97 


8 


1278 


1375 


1472 


1509 


1666 


1762 


1859 


1956 


2053 


2150 


97 


9 


2246 


2343 


2440 


2536 


2633 


2730 


2876 


2923 


3019 


3110 


97 


150 


653213 


653309 


653405 


653502 


653598 


053695 


653791 


653888 


653984 


654080 


96 


1 


4177 


4273 


4369 


4465 


4562 


4658 


4754 


4850 


4946 


5042 


96 


2 


5138 


5235 


5331 


5427 


5523 


5619 


5715 


5810 


5906 


6002 


96 


3 


6098 


6194 


6290 


6386 


6482 


6577 


6673 


6769 


6864 


6960 


96 


4 


7056 


7152 


7247 


7343 


7438 


7534 


7629 


7725 


7820 


7916 


m 


5 


8011 


8107 


8202 


8298 


8393 


8488 


8584 


«679 


8774 


8870 


93 


e 


8965 


906( 


9155 


9230 


9346 


9441 


9536 


9631 


9726 


9821 


95 


7 


9916 


650011 


630101 


C03231 


C6029G 


663331 


6G3486 


6G3581 


660676 


060771 


95 


h 


660865 


096U 


1055 


1150 


1245 


1339 


1434 


1529 


1623 


1718 


95 


s 


J813 


1907 


2002 


2396 


2191 


2286 


2380 


2475 


25C9 


2663 


95 



Wo. 10 111^131415161718 |9|D.a 



/ <)(r. U.' I TILVS OF NUMBERS. 



187 



NoJ 
1 

Q 
.1 

4 
5 
6 

7 



470 
1 
2 
3 
4 
5 
6 
7 
8 
9 

480 
1 
2 
3 
4 
5 
C 
7 
8 
9 

490 
1 
2 
3 



I 



I '^ I 4 



5 I 6 



r I 8 



9 I Diff 



002758 
3701 
4042 
558J 
0518 
7453 
8380 
9317 

670240 
1173 

072098 
3021 
3942 
4801 
5778 
6094 
7007 
8518 
9428 

680330 

081241 
2145 
3047 
3947 
4845 
574 
6036 
7529 
8420 
9309 

690196 
1081 
1905 
284- 
372' 
4005 
5482 
6356 
7229 
81U1 

098970 
9838 

700704 
1508 
2<i31 
3291 
4151 
5008 
5804 
0718 



i2852 
3795 
4736 
5075 
6612 
7540 
8479 
9410 
0339 
1205 

672190 
3113 
4034 
4953 
5870 
6785 
7698 
8009 
9519 

68U42G 

681332 
2235 
3137 
4037 
4935 
5831 
6720 
7018 
8509 
9398 

690285 
1170 
2053 
2935 
3815 
4693 
5569 
6444 
7317 
8188 



002947 
3HS9 
4830 
5709 
0705 
7040 
8572 
95(/3 

670431 
1358 

2283 
3205 
4120 

5045 
59r,2 
6876 
7789 
8700 
9610 
080517 

081422 
2320 
3227 
4127 

5025 
5921 
0815 
7707 
8598 
9480 

690373 
1258 
2142 
3023 
3903 
4781 
5057 
0531 
7404 
8275 



603041 063135 663230i663324 003418 603512 6636(r 



099057 099144 



r07570 
8421 
9270 

?ioir 

09g: 

1&07 
2650 

:->49i 

4330 
5167 



9924 
700790 
1054 
2517 
3377 
4230 
5094 
5949 
080j 

707055 
8500 
9355 

710202 
1048 
1892 
2734 
3575 
4414 
5251 



700011 
0877 
1741 
2603 
34()3 
4322 
5179 
0035 
0888 

707740 
8591 
9440 

71028 
li32 
1970 
2818 
3i;59 
4497 
5335 



3983 
4924 
58(52 
0799 
7733 
8005 
9590 
()70524 
1451 

672375 
3297 

42J8 
5137 
6053 
09C8 
7881 
8791 
9700 
680607 

681513 
24 K) 
3317 
4217 
5114 
6010 
6904 
7790 
8C8 
9575 

690402 
134' 
2230 
3111 
3991 
4808 
5744 

01 ;i 

7491 
8302 

099231 
700098 
09()3 
1827 
2089 
3549 
4408 
5205 
0120 
0974 



670017 670710 
15431 1636 



4078 
5018 
5950 
6892 
7826 
8759 
9089 



4172 
5112 

0050 
0980 
7920 

8852 
9782 



G72407 
3390 
4310 

5228 
0145 
7059 
7972 
8882 
9791 
080098 

681003 
2500 
3407 
4307 
5204 
0100 
6994 
788tj 
8770 
9004 

090550 
1435 
2318 
3199 

4078 
4950 
5832 
t)700 

7578 
8449 

09931 
700184 
1050 
1913 
2775 
3035 
4494 
5350 
0200 
7059 



ro7820 
8()70 
9524 

r 10371 
1217 
2000 
2902 
3742 
4581 
5418 



707911 
8701 
9009 

710456 
1301 
2144 
2980 
3820 
4605 
5502 1 



426() 4300 

5206 5299 

0143 6237 

7079 7173 

80131 8100 

8945' 9038 

98751 9j67 
070802 070895 

1728 i 1821 



672560 
3482 
4402 
5320 
6236 
7151 
8063 
8973 
9882 

680789 

681693 
2596 
3497 
4396 
5294 
0189 
7083 
7975 
8805 
9753 

09U()39| 
1524 
2400 
3287 
4100 
."044 

0793 
7665 
8535 

099404 
700271 
1130 
19J9 
28ol 
3721 
4579 
5430 
0291 
7144 

707996 
8846 
9t;94 

710540 
1385 
2229 
3070 
3910 
4749 
5586 



672652' 
35741 

4494 
54121 
6328 
7242 
8154 
9064 
9973 
680879 

681784 
2086 
3587 
4480 
5383 
0279 
7172 
8004 
8953 
9841 

69072?^ 
1012 
2494 
3375 
42.'>4 

Ot;t)7 

()880 
7752 
8022 

099491 
00358 
1222 
20H0 
2947 
3ci07 
4005 
5522 
037() 
7229 

r03081 
8931 
9779 

r 10025 
1470 
2313 
3154 
3994 
4833 
5669 



672744 
3006 
4586 
5503 
6419 
7333 
8245 
9155 

680003 
0970 



4454 
5.393 
0331 

7206 
8199 
913]! 
070000; 070 153! 
0988 108!) 
1913 2o05 93 



4548 94 

5487, 94 

6424 j 94 

7300 91 

8293 93 

9'224i 93 

53 j 93 

180 ! 93 



3077 
4570 
5473 
0308 
7201 
8153 
9042 
9930 

090810 
1700 
2583 
3403 
4312 
.">219 
0094 
0968 
7839 
8709 

099578 

00444 

1309 

2172 

3033 

;)d93 

4751 
5007 
0402 
7315 

r08l06 
9015 
9803 

^0710 

i.-.:.4l 
2:i.;| 

3238 
40781 
49161 
5753 



672836 
3758 
4077 
5595 
6511 
7424 
8336 
9246 

680154 
1000 

681904 
2807 
3707 
4G00 
5563 
0458 
7351 
8242 
9131 

690019 

690905 
1789 
2071 
3551 
4430 
530' 
6182 
7055 
7926 
8796 

699604 
700531 
1395 
2258 
,3119 
3979 
483 
5093 
0547 
7400 



708251 
9100 
9948 

710794 
1039 
2481 
3323 
4 J ('2 
5000 
5836 



672929 
3850 
4709 
5687 
6002 
7510 
8427 
9337 

080245 
1151 

682055 
2957 
3857 
4750 
5652 
0547 
7440 
8331 
9220 

69010' 

690993 
1877 
2759 
3039 
4517 
5394 
6209 
7142 
8014 
888:^ 

099751 
70001 
1482 
2344 
3205 
4005 
4922 
5778 
6632 
7485 

708336 
9185 

710033 
0879 
1723 
2566 
3407 
4246 

I 50»4 

I 5920 



No.| O 1 .1 1 ^ 1 3 1 4: 1 5 I 6 I 7 I 



84 
84 

I 9 \om 



188 



LOGARITHMS OF NIUIBERS. 



No. I O I 1 I 3 ( 



1*15 1 



I r I 8 f 



I DiS. 



520 


716003 


716087 716170 


716254 


716337 


716421 


716504 


716588 


716671 


716754 


83 


1 


6838 


6921 


7004 


7088 


7171 


7254 


7338 


7421 


7504 


7587 


83 


2 


7671 


7754 


7837 


7i>20 


8003 


8080 


81(59 


8253 


8336 


8419 


83 


3 


8502 


8585 


8668 


8751 


88.14 


8917 


9000 


90«3 


9165 


9248 


83 


4 


9331 


9414 


9497 


9580 


9()03 


9745 


9828 


9911 


9994 


720077 


83 


5 


720159 


720242 


720325 


720407 


720490 


720573 


720655 


720738 


720821 


0903 


83 


6 


0986 


1068 


1151 


1233 


1316 


1398 


1481 


1563 


1046 


1728 


82 


7 


1811 


1893 


1975 


2058 


2140 


Q-)00 


2305 


2387 


2469 


2552 


82 


8 


2034 


2716 


2798 


2881 


2963 


3045 


3127 


320:) 


3291 


3374 


82 


9 


3456 


3538 


3620 


3702 


3784 


3800 


3948 


4030 


4112 


4194 


82 


530 


724276 


724358 721440 


724522 


724604 


724085 


724767 


724849 


724931 


725013 


82 


1 


5095 


5176 


5258 


5340 


5422 


5503 


5585 


5667 


5748 


5830 


82 


2 


5912 


5993 


6075 


6156 


6238 


6320 


6401 


6483 


6564 


6(546 


82 


3 


6727 


6809 


6890 


6972 


7053 


7134 


7216 


7297 


7379 


7460 


81 


.4 


7541 


7023 


7704 


7785 


7860 


7948 


8029 


8110 


8191 


8273 


81 


5 


8354 


8435 


8516 


8597 


8678 


8759 


8841 


8922 


9003 


9084 


81 


6 


9165 


9240 


9327 


9408 


9489 


9570 


9651 


9732 


9813 


9893 


81 


7 


9974 


730055 


730136 


730217 


730298 


730378 


730459 


730540 


730621 


730702 


81 


8 


730782 


0803 


0944 


1024 


1105 


1186 


1266 


1347 


1428 


1508 


81 


9 


158D 


1869 


1750 


1830 


1911 


1991 


2072 


2152 


2233 


2313 


81 


540 


732394 


732474 


732555 


732635 


732715 


732796 


732376 


732956 


733037 


733117 


80 


1 


3197 


3278 


3358 


3433 


3518 


3598 


3679 


3759 


3839 


3919 


80 


2 


3999 


4079 


4160 


4240 


4320 


4400 


4480 


4560 


4640 


4720 


80 


3 


4800 


4880 


4960 


5040 


5120 


5200 


5279 


5359 


5439 


5519 


80 


4 


5599 


5679 


5759 


5;S3^ 


5918 


5998 


(5078 


6157 


6237 


(5317 


80 


5 


6397 


6476 


0556 


6035 


6715 


6795 


0874 


6954 


7034 


7113 


80 


6 


7193 


7272 


7352 


743! 


7511 


7590 


7670 


7749 


7829 


7908 


79 


7 


7987 


8067 


8140 


822') 


8305 


8384 


8403 


8543 


8622 


8701 


79 


8 


8781 


8860 


8939 


9!)!H 


9097 


9177 


9256 


9335 


9414 


9493 


79 


9 


9572 


9651 


9731 


9810 


9889 


9968 


740047 


740126 


740205 


740284 


79 


550 


740363 


740442 


740521 


740000 


740078 


740757 


740836 


740915 


740994 


741073 


79 


1 


1152 


1230 


1309 


1388 


1467 


1540 


1624 


1703 


1782 


1860 


79 


o 


1939 


2018 


2096 


2175 


2254 


2332 


2411 


2489 


2568 


2047 


79 


3 


2725 


2804 


2382 


29(U 


3039 


3118 


3196 


3275 


3353 


3431 


78 


4 


3510 


3588 


3607 


3745 


3823 


3902 


3980 


4058 


4136 


4215 


78 


5 


4293 


4371 


4449 


4528 


4600 


4684 


4762 


4840 


4919 


4997 


78 


6 


5075 


5153 


5231 


5309 


5387 


5405 


5543 


5621 


5699 


5777 


78 


7 


5855 


5933 


6011 


6089 


6167 


6245 


0323 


6401 


6479 


6556 


78 


8 


6634 


6712 


6790 


68()8 


6945 


7023 


7101 


7179 


7256 


7334 


78 


9 


7412 


7489 


7567 


7645 


7722 


7800 


7878 


7955 


8033 


8110 


78 


5G0 


748188 


748266 


748343 


748421 


748498 


748576 


748(553 


748731 


748808 


748885 


77 


] 


8963 


9040 


9118 


9195 


9272 


9350 


9427 


9504 


9582 


9059 


77 


2 


9736 


9814 


9891 


9968 


750045 


750123 


750200 


750277 


750354 


750431 


77 


3 


750508 


750586 


750663 


750740 


0817 


0894 


0971 


1048 


1125 


1202 


77 


4 


1279 


1356 


1433 


• 1510 


1587 


1604 


1741 


1818 


1895 


1972 


77 


5 


2048 


2125 


2202 


2279 


2350 


2433 


2509 


2586 


2663 


2740 


77 


6 


2816 


2893 


2970 


3047 


3123 


3200 


3277 


3353 


3430 


3500 


77 


7 


3583 


3060 


3736 


3813 


3889 


39156 


4042 


4119 


4195 


4272 


77 


8 


434S 


4425 


4501 


4578 


4654 


4730 


4807 


4883 


4900 


503(5 


76 


9 


5112 


5189 


5265 


5341 


5417 


5494 


5570 


5646 


5722 


5799 


76 


no 


755875 


755951 


750027 


750103 


756180 


756256 


756332 


750408 


756484 


75(5560 


76 


I 


6636 


6712 


6788 


6804 


6940 


7016 


7092 


7168 


7244 


7320 


76 


2 


7396 


7472 


7548 


7624 


7700 


7775 


7851 


7v>27 


8003 


8079 


76 


3 


815S 


8230 


830(5 


8382 


8458 


8533 


8009 


8685 


8761 


8836 


76 


4 


8912 


8988 


9063 


9139 


9214 


9290 


9366 


9441 


9517 


9592 


76 


5 


9668 


9743 


9819 


9894 


9970 


760045 


760121 


760196 


760272 


760347 


75 


6 


760422 


760498 


760573 


7011049 


7(50724 


0799 


0875 


0950 


1025 


1101 


75 


7 


1176 


1251 


132(i 


14()J 


1477 


1552 


1()27 


1702 


1778 


1853 


75 


8 


1928 


2003 


2078 


2153 


2228 


2303 


2378 


2453 


2529 


2604 


75 


9 


2679 


2754 


2829 


2904 


2978 


3053 


3128 


3203 


3278 


3353 


75 



No.|0|l|S|3|4:|5|6|7|8|9|Diff 



LOGARITHMS OF NUMBERS. 



189 



No. 


i 


1 1 


1 3 


1 3 


i * 


1 5 


1 6 


7 


8 


9 


iDiff 


580 


763428 


763503 


763578 


763653 


763727 


763802 


763877 


763952 


764027 


764101 


75 


1 


4176 


4251 


4326 


4400 


4475 


4550 


4624 


4699 


4774 


4848 


75 


o 


4923 


4998 


5072 


5147 


5221 


5296 


5370 


5445 


5520 


5594 


75 


3 


5669 


5743 


5818 


5892 


5966 


6041 


6115 


6190 


6264 


6338 


74 


4 


6413 


(>487 


6562 


6G36 


6710 


6785 


GS59 


0933 


7007 


7082 


74 


5 


7156 


7230 


7304 


7379 


7453 


7527 


7601 


7075 


7749 


7823 


74 


f) 


7898 


7972 


8046 


8120 


8194 


8268 


8342 


8410 


8490 


8564 


74 


7 


8638 


8712 


87o6 


88G0 


8934 


9008 


9082 


9156 


9230 


9303 


74 


8 


9377 


9451 


9525 


9599 


9G73 


9746 


9820 


9894 


9968 


770042 


74 


9 


770115 


770189 


770263 


770336 


770410 


770484 


770557 


770631 


770705 


0778 


74 


590 


770852 


770926 


770999 


771073 


771146 


771220 


771293 


771307 


771440 


771514 


74 


1 


1587 


1661 


1734 


1808 


1881 


1955 


2028 


2102 


2175 


2248 


73 


2 


2322 


2395 


24G8 


2542 


2615 


2688 


270x,M 2835 


2908 


2981 


73 


3 


3055 


3128 


3201 


3274 


3348 


3421 


3494; 3567 


3040 


3713 


73 


4 


3786 


3860 


3933 


4006 


4079 


4152 


4225 


4293 


4371 


4444 


. 73 


5 


4517 


4590 


46iJ3 


4736 


4809 


4882 


4955 


5G28 


5100 


5173 


73 


6 


5246 


5319 


5392 


5465 


5538 


5G10 


5683 


5756 


5829 


5902 


73 


7 


5974 


6047 


6123 


6193 


62G5 


6338 


6411 


6483 


6556 


6629 


73 


8 


6701 


6774 


G84G 


6919 


6992 


7084 


7137 


7209 


7282 


7354 


73 


9 


7427 


7499 


7572 


7644 


7717 


7789 


78G2 


7934 


8006 


8079 


72 


600 


778151 


778224 


778290 


778368 


778441 


778513 


778585 


778658 


778730 


778802 


72 


1 


8874 


8947 


9019 


9091 


9163 


9236 


9308 


9380 


9452 


9524 


72 


2 


9596 


9069 


9741 


9813 


9885 


9957 


780029 


783101 


783173 


780245 


72 


3 


780317 


780389 


780461 


780533 


780605 


780877 


0749 


0821 


0833 


0965 


72 


4 


1037 


1109 


1181 


1253 


1324 


1396 


1468 


1540 


1612 


1684 


72 


5 


1755 


1827 


1839 


1971 


2042 


2114 


21 8G 


2258 


2329 


2401 


72 


6 


2-173 


2544 


2G1G 


2688 


2759 


2831 


2902 


2974 


304G 


3117 


72 


7 


3189 


3260 


3332 


3403 


3475 


3546 


3GI8 


3689 


37G1 


3832 


71 


8 


3904 


3975 


4046 


4118 


4189 


4261 


4333 


4403 


4475 


4546 


71 


9 


4617 


4C89 


4760 


4831 


4932 


4974 


5045 


5116 


5187 


5259 


71 


610 


785330 


785401 


785472 


785543 


785615 


785686 


785757 


785828 


785899 


785970 


71 


1 


6;)41 


6112 


6183 


6254 


6325 


6393 


C467 


6538 


6609 


6680 


71 


2 


0751 


0822 


6893 


6964 


7035 


7103 


7177 


7248 


7319 


7390 


71 


3 


74G3 


7331 


7602 


7673 


7744 


7815 


7885 


7956 


8027 


8098 


71 


4 


8168 


8233 


8310 


8381 


8451 


8522 


8593 


8G63 


8734 


8804 


71 


5 


8375 


8346 


9016 


9U87 


9157 


9228 


9239 


9369 


9440 


9510 


71 


6 


9581 


9G51 


9700 


9792 


9863 


9933 


790004 


790074 


790144 


790215 


70 


7 


790235 


790356 


790426 


790496 


790567 


790637 


0707 


0778 


0848 


0918 


70 


8 


0988 


1059 


1129 


1193 


1269 


13-10 


1410 


1480 


1550 


1620 


70 


9 


1631 


1761 


1831 


1901 


1971 


2341 


2111 


2181 


2252 


2322 


70 


620 


792393 


792462 


792532 


792602 


792672 


792742 


792812 


792882 


792952 


793022 


70 


1 


3092 


3162 


3231 


330] 


3371 


3-141 


3511 


3581 


3651 


3721 


70 


2 


3793 


3360 


3930 


4000 


4070 


4139 


4209 


4279 


4349 


4418 


70 


3 


4488 


4558 


4627 


4697 


4767 


4836 


4996 


4976 


5045 


5115 


70 


4 


5185 


5254 


5324 


5393 


5463 


5532 


5602 


5672 


5741 


5811 


70 


5 


5883 


5949 


6019 


6088 


6158 


6227 


6297 


6366 


6436 


6505 


69 


6 


6574 


6644 


6713 


6782 


6852 


6921 


6990 


7060 


7129 


7198 


69 


7 


7268 


7337 


74G0 


7475 


7545 


7614 


7G83 


7752 


7821 


7890 


69 


8 


7960 


8029 


8098 


8167 


8236 


8305 


8374 


8443 


8513 


8582 


69 


9 


8651 


8720 


8789 


8858 


8927 


8930 


90G5 


9134 


9203 


9272 


69 


330 


799341 


799409 


799478 


799547 


799616 


793685 


799754 


799823 


799892 


799961 


69 


1 


800029 


800098 


800167 


800236 


800305 


800373 


800442 


800511 


800580 


800648 


69 


2 


0717 


0786 


0854 


0923 


09G2 


ICGI 


1123 


1198 


1266 


1335 


69 


3 


1404 


1472 


1541 


1609 


1678 


1747 


1815 


1884 


1952 


2021 


69 


4 


2089 


2158 


2226 


2295 


2363 


2433 


2500 


2563 


2637 


2705 


68 


5 


2774 


2842 


2910 


2979 


3047 


31JG 


3184 


3252 


3321 


3389 


68 


G 


3457 


3525 


3594 


3GG2 


3730 


3793 


3867 1 3935 


4003 


4071 


68 


7 


4139 


4208 


4276 


4344 


4412 


4480 


45481 4616 


4685 


4753 


68 


8 


4821 


4889 


4957 


5025 


5093 


5161 


52391 5297 


5365 


5433 


68 


& 


5501 


5569 


5637 


5705 


5773 


58411 59081 5976 


6044 


6112 


68 



No. I 



|lia|3|4:|5|6|7|8|9|D^ 



190 



LOGARITHMS OF NUMBERS. 



N* 


1 


1 1 


3 


3 


4 


1 5 


1 6 


7 


1 8 


9 


I Did 


t)4(l 


8()(;iHn 


806^^l8:8063]6iS06384 


806451 


806519 


806587 


806655 


806-'23 


8067901 68 
74671 68 


J 


0858 


f.!)26 


6994^ 7061 


7129 


7197 


72(54 


7332 


7400 


o 


7535 


7(503 


7670 


7738 


7806 


7873 


7941 


8008 


8076 


8143 


68 


3 


8211 


8279 


8346 


8414 


8481 


8549 


861(5 


8(584 


8751 


8818 


67 


4 


8886 


8953 


9021 


9088 


9156 


9223 


9290 


9358 


9425 


9492 


67 


."5 


9560 


9627 


9HH4 


9762 


9829 


9896 


f)l)(54 


810031 


810098 


8101(5S 


67 


6 


810233 


810300 


810367 


810434 


810501 


810569 


810(536 


0703 


0770 


0837 


67 


7 


0904 


0971 


1039 


110(5 


1173 


1240 


1307 


1374 


1441 


1508 


(57 


8 


J 575 


1642 


1709 


1776 


1«43 


19 "0 


l;/77 


2044 


2111 


2178 


67 


9 


2245 


2312 


2379 


2445 


2512 


2579 


2(54(5 


2713 


2780 


2847 


67 


650 


8] 29 13 


812980 


813047 


813114 


813181 


813247 


813314 


813381 


813448 


813514 


67 


i 


3581 


3648 


3714 


3781 


3848 


3914 


39«1 


4048 


4114 


4181 


67 


2 


4J48 


4314 


4381 


4447 


4514 


4581 


4647 


i-lW 


4780 


4847 


67 


3 


4913 


4980 


5046 


5113 


5179 


5246 


5312 


5378 


5445 


5511 


66 


4 


5578 


5644 


5711 


5777 


5843 


5910 


5976 


6042 


6109 


6175 


66 


5 


6241 


6308 


6374 


GMO 


6503 


6573 


6G39 


6705 


6771 


G838 


6(5 


6 


0904 


0970 


7036 


7102 


7169 


7235 


7301 


7367 


7433 


7499 


66 


7 


75G5 


7631 


7698 


7764 


7830 


7896 


79G2 


80;:8 


8094 


8160 


66 


8 


8226 


8292 


83.^8 


8^:^ 


8490 


8556 


8622 


8688 


8754 


8820 


66 


9 


8885 


8951 


9917 


9083 


9149 


9215 


9281 


9346 


9412 


9478 


66 


660 


819544 


8I9G10 


8IC676 


819741 


819807 


819873 


819939 


820004 


820070 


820136 


06 


1 


820201 


820267 


820333 


820399 


820464 


820530 


820595 


0661 


6727 


0792 


66 


2 


0858 


09-34 


09!^9 


105,3 


1120 


1186 


1251 


1317 


1382 


1443 


66 


3 


1514 


1579 


1645 


1710 


1775 


1841 


1906 


1972 


2337 


2103 


05 


4 


2168 


2233 


2299 


2364 


2430 


2495 


2560 


2626 


2691 


2756 


65 


5 


2822 


2887 


2952 


3018 


3083 


3148 


3213 


3279 


3344 


3409 


65 


6 


3474 


3539 


3i5i:.-, 


3i570 


3735 


3800 


3335 


393,) 


3996 


4061 


65 


7 


4126 


4191 


425'5 


4321 


4386 


4451 


4516 


4581 


4646 


4711 


65 


8 


4776 


4841 


49!)G 


4971 


503G 


5101 


51G6 


3231 


5296 


5361 


65 


9 


5426 


5491 


555t5 


5621 


5686 


5751 


5815 


5880 


5945 


6010 


65 


670 


826075 


826140 


826204 


826269 


826334 


820399 


826464 


826528 


826593 


826658 


65 


1 


6723 


6787 


6852 


6917 


6981 


7046 


7111 


7175 


7240 


7305 


65 


2 


7369 


7434 


7499 


75G3 


7G28 


7692 


7757 


7821 


7886 


7951 


65 


3 


8015 


8080 


8144 


8209 


8273 


8338 


8-102 


8467 


8531 


8595 


64 


4 


8660 


8724 


8789 


8853 


8918 


8982 


9346 


9111 


9175 


9239 


64 


5 


9304 


ur.np 


9:'^2 


9497 


9561 


9625 


9GJ0 


9754 


9818 


9882 


64 


6 


9947 


83001! 


830075 


830139 


830204 


83026a 


83u332 


83039(5 


830460 


830525 


64 


7 


830589 


o(i:.3 


(:717 


C781 


0845 


0909 


6973 


1037 


1102 


1166 


64 


8 


1230 


1294 


1358 


1422 


1480 


1550 


1614 


1678 


1742 


1806 


64 


9 


1870 


19;!4 


1998 


2062 


2126 


2189 


2253 


2317 


2381 


2445 


64 


680 


832509 


83257:; 


8;!2;^57 


832700 


832764 


832828 


832892 


832956 


833920 


833083 


64 


J 


3147 


3211 


3:75 


3338 


3402 


3466 


3530 


3593 


3657 


3721 


64 


2 


3764 3848 


3i)12 


3975 


4039 


4103 


4166 


4230 


4294 


4357 


64 


3 


4421 


4484 


4548 


4611 


4675 


4739 


4802 


4866 


4929 


4993 


64 


4 


5056 


512U 


5183 


5247 


5310 


5373 


5437 


5500 


5564 


5627 


63 


5 


5691 


5754 


5817 


5881 


5944 


6007 


6071 


6134 


6197 


6261 


63 


6 


6324 


6337 


0451 


6514 


6577 


6641 


6704 


6767 


6830 


6894 


63 


7 


6957 


7020 


7083 


7146 


7210 


7273 


733(5 


7399 


7462 


7525 


63 


8 


7588 


7652 


7715 




7841 


7904 


7967 


8030 


8093 


8156 


63 


9 


8219 


8282 


8345 


8408 


8471 


8534 


8597 


8660 


8723 


8786 


63 


690 


838849 


S38912 


838975 


839038 


839101 


839164 


839227 


839289 


839352 


839415 


63 


1 


9478 


9541 


9604 


9(567 


9729 


9792 


9855 


9918 


9981 


840043 


63 


2 


8401)6 


84016!) 


840232 


840294 


840357 


840420 


840482 


840545 


840(508 


0671 


63 


3 


0733 


0793 


0859 


0921 


0984 


1046 


1109 


1172 


1234 


1297 


63 


4 


1359 


1422 


1485 


1547 


1610 


1672 


1735 


1797 


1860 


1922 


63 


5 


1985 


2047 


2110 


2172 


2235 


2297 


2360 


2422 


2484 


2.547 


62 


6 


2609 


2(572 


2734 


2796 


2859 


2921 


2983 


3046 


3108 


3170 


62 


7 


3233 


3295 


3357 


3420 


3482 


3544 


3606 


3(569 


3731 


3793 


62 


8 


3855 3918 


3980 


4042 


4104 


4166 


4229 


4201 


4353 


4415 


62 


9 


4477 4539 


4601 


4664 


4726 


4788 


4850 


4912 


4974 


5036 


62 



No. I O I 1 



I 3 I 4 I 5 j 6 I 7 I 8 I 9 |D>£ 



LOGARITHMS OF NUlfBERS. 



19] 



No. I 



I 3 



I * I 



8 I 9 I Diff. 



"700 


845098 


845160 


845222 


845284 


845346 


84.5408 


845470 


845532 


845594 


845656 


62 


] 


57J8 


5780 


5842 


5904 


5966 


6028 


6090 


6151 


6213 


6275 


62 


2 


6337 


6399 


6461 


6523 


6585 


6646 


6708 


6770 


6832 


6894 


62 


3 


6955 


7017 


7079 


7141 


7202 


7264 


7326 


7388 


7449 


7511 


62 


4 


7573 


7634 


7696 


7758 


7819 


7881 


7943 


8004 


8066 


8128 


62 


5 


8189 


8251 


8312 


8374 


8435 


8497 


8559 


8620 


8682 


8743 


62 


6 


8805 


8866 


8928 


8980 


9051 


9112 


9174 


9235 


9297 


9358 


61 


'/ 


9419 


9481 


9.542 


9(504 


9665 


9720 


9788 


9849 


9911 


9972 


61 


8 


850033 


850005 


850156 


850217 


850279 


850340 


850401 


850462 


850524 


850585 


61 


9 


064!) 


0707 j 076^) 


()83(» 


0891 


0952 


1014 


1075 


1136 


1197 


61 


710 


851258 


851320 


851381 


851442 


851503 


851564 


851625 


851686 


851747 


851809 


61 


1 


1870 


1931 


1992 


2053 


2114 


2175 


2236 


2297 


2358 


2419 


61 


2 


2480 


2541 


2602 


2663 


0704 


2785 


2846 


2907 


2968 


3029 


61 


3 


3090 


3150 


3211 


3272 


3333 


3394 


3455 


3516 


3577 


3637 


61 


4 


3698 


3759 


3820 


3881 


3941 


4002 


4063 


4124 


4185 


4245 


61 


5 


4306 


4367 


4428 


4488 


4549 


4610 


4670 


4731 


4792 


4852 


61 


6 


4913 


4974 


5034 


5095 


^156 


5216 


5277 


5337 


5398 


5459 


61 


7 


5519 


5580 


5640 


5701 


5761 


5822 


5882 


5943 


6003 


6064 


61 


8 


611M 


6185 


6245 


6306 


6366 


6427 


6487 


6548 


6608 


6668 


60 


9 


6729 


6789 


6850 


6910 


6970 


7031 


7091 


7152 


7212 


7272 


60 


"20 


857332 


857393 


8574 j3 


857513 


357574 


857534 


857694 


857755 


857815 


857875 


60 


1 


7935 


7995 


8056 


8116 


8176 


8236 


8297 


8357 


8417 


8477 


60 


2 


8537 


8597 


8657 


8718 


8778 


8838 


8898 


8958 


9018 


9078 


60 


3 


9138 


9198 


9258 


9318 


•9379 


9439 


9499 


9559 


9619 


9679 


60 


4 


9739 


97U9 
860398 


9859 


9918 


9978 


860038 


860098 


860158 


860218 


860278 


60 


5 


860338 


800158 


860518 


860578 


0637 


0697 


0757 


0817 


0877 


60 


6 


0937 


0996 


1056 


1116 


1176 


1236 


1295 


1355 


1415 


1475 


60 




1534 


1594 


1654 


1714 


1773 


1833 


1893 


1952 


2012 


2072 


60 


8 


2131 


2191 


2251 


2310 


2370 


2430 


2489 


2549 


2608 


2668 


60 


9 


2728 


2787 


2847 


2906 


2966 


3025 


3085 


3144 


3204 


3263 


60 


730 


863323 


863382 


663442 


863501 


863561 


863620 


863680 


863739 


863799 


863858 


59 


i 


3917 


3977 


4036 


4096 


4155 


4214 


4274 


4333 


4392 


4452 


59 


2 


4511 


4570 


4630 


4r89 


4748 


4808 


4867 


4926 


4985 


5045 


59 


3 


5104 


5163 


5222 


5282 


5341 


54(10 


5459 


5519 


5578 


5637 


59 


4 


5696 


5755 


5814 


5874 


5933 


5992 


6051 


6J10 


6169 


6228 


59 


5 


6287 


6346 


6405 


6465 


6524 


6583 


6642 


6701 


6760 


6819 


59 


6 


6878 


6937 


6996 


7055 


7114 


7173 


7232 


7291 


7350 


7409 


59 




7467 


7526 


7585 


7644 


7703 


7762 


7821 


7^80 


7939 


7998 


59 


8 


8056 


8115 


8174 


8233 


8292 


835U 


8409 


8468 


8527 


8586 


59 


9 


8&44 


8703 


8762 


8821 


8879 


8938 


8997 


9056 


9114 


9173 


59 


740 


809232 


8'J9290 


869349 


869408 


869486 


869525 


869584 


869642 


869701 


869760 


59 


1 


9818 


9877 


9935 


9994 


870053 


870111 


870170 


870228 


870287 


870345 


59 


•2 


870404 


8704O2 


870521 


370579 


0638 


0696 


0755 


0813 


0872 


0930 


58 


3 


0989 


IU47 


1106 


1164 


1223 


1281 


1339 


1398 


1456 


1515 


58 


4 


1573 


1631 


1690 


1748 


1806 


1865 


li)23 


1981 


2040 


2098 


58 


5 


2lo6 


2215 


2273 


2331 


2389 


2448 


2506 


2564 


2622 


2681 


58 


6 


2739 


2797 


2855 


2913 


2972 


3030 


3088 


3146 


3204 


3262 


58 


7 


3321 


3379 


3437 


3495 


3553 


3611 


3669 


3727 


3785 


3844 


58 


8 


3902 


3960 


4018 


4076 


4134 


4192 


4250 


4308 


4366 


4424 


58 


9 


4482 


4540 


4598 


4656 


4714 


477g 


4830 


4888 


4945 


5003 


58 


750 


875061 


875: 19 


875177 


875235 


875293 


875351 


875409 


875466 


875524 


875582 


58 


1 


5640 


5698 


5756 


5813 


5871 


5929 


5987 


6045 


6102 


6160 


58 


2 


6218 


6276 


6333 


6391 


6449 


6507 


6564 


6622 


6680 


6737 


58 


3 


6795 


6853 


6010 


6968 


7026 


7083 


7141 


7199 


7256 


7314 


58 


4 


7371 


7429 


7487 


7544 


7602 


7659 


7717 




7832 


7889 


58 


5 


7947 


8004 


8062 


8119 


8177 


8234 


8292 


8349 


8407 


8464 


57 


6 


8522 


8579 


8637 


8694 


8752 


8809 


8866 


8924 


8981 


9039 


57 


7 


9090 


9153 


9211 


9268 


9325 


9383 


9440 


9497 


9555 


9612 


57 


8 


9S09 


9726 


9784 


9841 


9898 


9956 


380013 


880070 


880127 880185] 


57 


9; 880-242 


88C299I 


8803561 


8804131 


880471 


88052tf 


0585 


0642 


06991 07561 


57 



N*lO|l|3|3|4:j5j 



I 7 I 8 i 9 |Di£ 



192 



LOGARITHMS OF NlUfBEBS, 



No 





1 


3 


3 


4 


5 





7 


8 


9 


Dift 


7fX) 


880814 


H8087] 


880..28| 880985 


881042 


881099 


881156 


881213,881271 


8813-28 


~;>7 


1 


1385 


1442 


149it 


1556 


1613 


1670 


17-27 


1784 


1841 


1898 


5' 


2 


1955 


2012 


2069 


2126 


2183 


2240 


2297 


2354 


2411 


2468 


57 


3 


2525 


2581 


2()38 


2r>95 


2752 


28i)9 


2866 


2923 


2980 


3037 


57 


4 


3093 


3150 


3207 


3264 


3321 


3377 


3434 


3491 


3548 


3605 


57 


5 


36(31 


3718 


3775 


3832 


3888 


3945 


4002 


4059 


4115 


4172 


57 


«) 


4-229 


4285 


4342 


4399 


4455 


4512 


4569 


4625 


4(i82 


4739 


57 


7 


47;).') 


4852 


4909 


4965 


5j22 


5078 


5135 


5192 


5243 


5305 


57 


8 


53()1 


5418 


5474 


5531 


558/ 


5644 


5700 


5757 


53i;{ 


5870 


57 


9 


5i)2t) 


5983 


6039 


6096 


6152 


6209 


6265 


6321 


6378 


6434 


56 


770 


886491 


888547 


886604 


886tu;o 


8S6716 


886773 


886829 


886885 


886942 


886998 


56 


1 


7054 


71J1 


7167 


7223 


7280 


7331) 


7392 


7449 


7505 


7561 


56 


2 


7617 


7674 


7730 


778(i 


7842 


7898 


7955 


8011 


8067 


8123 


56 


3 


8179 


8236 


8292 


8348 


8404 


84()0 


8516 


8573 


8629 


8685 


56 


4 


8741 


8797 


8853 


8999 


8965 


9021 


9077 


9134 


9190 


9246 


56 


5 


9302 


9358 


9414 


9470 


9526 


9582 


9633 


9694 


9750 


9806 


56 


6 


9862 


9918 


9974 


890030 


890086 


890141 


890197 


890253 


890309 


890365 


56 


7 


890421 


890477 


890533 


0589 


0(i45 


0700 


0756 


0812 


0868 


0921 


56 


S 


0980 


1035 


1091 


1147 


12:)3 


1259 


1314 


1370 


1426 


1482 


56 


9 


1537 


1593 


1649 


1705 


176U 


181.; 


1872 


1928 


1983 


2039 


56 


rso 


892095 


892150 


892296 


892262 


892317 


89237:-: 


892429 


892484 


892540 


892595 


56 


1 


2651 


2707 


2762 


2818 


2873 


2J2J 


29.-<5 


3040 


3096 


3151 


56 


2 


3207 


3262 


3318 


3373 


3429 


3484 


354;) 


3595 


3651 


3706 


56 


3 


3762 


3dlV 


3873 


3928 


3984 


4039 


40J1 


4150 


4205 


4261 


55 


4 


4316 


4371 


4427 


4482 


4533 


4593 


4648 


4704 


4759 


4814 


55 


5 


4870 


4925 


4980 


5;i;j6 


5091 


5146 


5201 


5257 


5312 


5367 


55 


6 


5423 


5478 


5533 


5588 


5644 


5699 


5754 


5809 


5864 


5920 


55 


7 


5975 


6U30 


6085 


6140 


6195 


6251 


6306 


6361 


6416 


6471 


55 


8 


6526 


658.1 


6636 


6692 


6747 


6802 


6857 


6912 


6967 


7022 


55 


9 


7077 


7132 


7187 


7212 


7297 


7352 


7407 


7462 


7517 


7572 


55 


790 


897627 


897682 


897737 


897792 


897847 


897902 


897957 


898012 


898067 


898122 


55 


1 


8176 


8231 


8286 


8341 


8396 


8451 


850() 


8561 


8()15 


8670 


55 


2 


8725 


8780 


8835 


8890 


8944 


8999 


9054 


9109 


9164 


9218 


55 


3 


9273 


9328 


9383 


9437 


9492 


9547 


9602 


965() 


9711 


9766 


55 


4 


9821 


9875 


9930 


9985 


900039 


900094 


900149 


900203 


900258 


900312 


55 


J 


900367 


900422 


90U476 


900531 


0586 


0640 


0695 


0749 


0804 


0859 


55 


6 


0913 


0968 


1022 


1077 


1131 


118;; 


1240 


1295 


1349 


1404 


55 


7 


1458 


1513 


1567 


1622 


1676 


1731 


1785 


1840 


1894 


1948 


54 


8 


2003 


2057 


2112 


2166 


2221 


2-275 


2329 


2384 


2438 


2492 


54 


9 


2547 


2601 


2655 


2710 


2764 


2818 


2873 


2927 


2981 


3036 


54 


800 


903090 


9J3I44 


903199 


903253 


903307 


903361 


903416 


903470 


903524 


903578 


54 


1 


3633 


3687 


3741 


3795 


3849 


3904 


3958 


4012 


4066 


4120 


54 


2 


4174 


4229 


4283 


4337 


4391 


4445 


44 9; > 


4553 


4607 


4661 


54 


3 


4716 


477.J 


4824 


4878 


4932 


4986 


5040 


5094 


5148 


5202 


54 


4 


5256 


5310 


53(M 


5418 


5472 


5526 


5530 


5634 


5688 


5742 


54 


5 


5796 


5850 


5904 


5958 


6012 


6066 


6119 


6173 


0-227 


6231 


54 


6 


6335 


6389 


6443 


6497 


6551 


6(i04 


6658 


6712 


6766 


6820 


54 


7 


6874 


6927 


6931 


7035 


7089 


7143 


7196 


7250 


7304 


7358 


54 


8 


7411 


74G5 


7519 


7573 


7621) 


7680 


7734 


7787 


7841 


7895 


51 


9 


7949 


8002 


8056 


8110 


8103 


8217 


8270 


8324 


8378 


8431 


54 


810 


908485 


908539 


908592 


908646 


908699 


908753 


908807 


908860 


908914 


908967 


54 


1 


9021 


9i)74 


9128 


9181 


9235 


9289 


9342 


9396 


9449 


9503 


54 


2 


955() 


yijio 


9663 


971 ;; 


9770 


9823 


9877 


9930 


9984 


910037 


53 


3 


910U91 


910144 


910197 


910251 


910304 


910358 


910411 


910464 


910518 


0571 


53 


4 


0624 


0673 


0731 


0784 


0838 


0891 


0944 


0998 


1051 


1104 


53 


5 


1158 


1211 


12(i4 


1317 


1371 


1424 


1477 


1530 


1584 


1637 


53 


6 


1690 


1743 


1797 


1850 


1903 


1956 


2009 


2963 


2116 


21G9 


53 


7 


2222 


2275 


2328 


2381 


2435 


2488 


2541 


2594 


2647 


2700 


53 


8 


2753 


2806 


2859 


2913 


2966 


3019 


3072 


31-25 


3178 


3231 


53 


9 


3284 


3337 


3390 


3443 


3496 


3549 


3602 


3655 


3708 


3761 


53 



3|4|5|6|7j8j9JDiff 



%1 



I-OGAIUTIUIS OF XUMBERS. 



193 



Na I I 1 



3 I 3 I 4 



7 I 8 



9 I 



820 


913814 


913867 913920 913973 


914026 


914079 


914l32i914184 


914237 


914290 


53 


1 


4343 


4396 


4449 


4502 


4555 


4608 


4660 


4713 


4766 


4819 


53 


2 


4872 


4925 


4977 


5030 


5083 


5136 


5189 


5241 


5294 


5347 


53 


3 


5400 


5453 


5505 


5558 


5611 


5664 


5716 


5769 


5822 


5875 


53 


4 


5927 


59S0 


6033 


6085 


6138 


6)91 


6243 


6296) 


6349 


6401 


53 


5 


6454 


6507 


6559 


6612 


6664 


6717 


6770 


6822 


6875 


6927 


53 


6 


. 6980 


7033 


7085 


7338 


7190 


7243 


7295 


7348 


7400 


7453 


53 


7 


7506 


75.58 


7611 


7663 


7716 


7768 


7820 


7873 


7925 


7978 


52 


8 


8030 


8083 


8135 


8188 


8240 


8293 


8345 


8397 


8450 


8502 


52 


9 


8555 


8607 


8t;59 


8712 


87(14 


8816 


8869 


8921 


8973 


9026 


52 


830 J9 19078 


919130 


919183 919235 


919287 


919340 


919392 


919444 


919496 


913549 


52 


1 


9601 


9653 


97(16 9758 


9810 


9862 


9914 


9967 


920019 


920071 


52 


2 


920123 


920176 


92U228 92l»-if'0 


920332 


520384 


9204361920489 


054 1 


0593 


52 


3 


0645 


0697 


074'.l 08U1 


0853 


0906 


09581 1010 


1062 


1114 


52 


4 


1166 


1218 


1270 


1322 


1374 


1426 


1478 


1530 


1582 


1634 


52 


5 


1686 


1738 


1790 


1842 


1894 


1946 


1998 


2050 


2102 


2154 


52 


6 


2206 


2258 


2310 


2362 


2414 


2466 


2516 


2570 


2622 


2674 


52 


7 


2725 


2777 


2829; 2881 


2933 


2985 


3037 


3089 


3140 


3192 


52 


8 


3244 


3296 


3348 


3399 


3451 


3503 


3555 


3607 


3658 


3710 


52 


9 


3762 


3814 


381)5 


3917 


3969 


4021 


4072 


4124 


4176 


4228 


52 


840 


924279 


924331 


924383 


924434 


924486:924538 


924589 


92464! 


924693 


924744 


52 


1 


4796 


4848 


4899 


4951 


5003 


5(S54 


5106 


5157 


5209 


5261 


52 


2 


5312 


5364 


5415 


5467 


5518 


5570 1 5621 


5673 


5725 


5776 


52 


3 


5828 


5879 


5931 


5982 


6034 


6085; 6137 


6188 


6240 


6291 


51 


4 


6342 


6394 


6445 


6497 


6548 


6600 


6651 


6702 


6754 


6805 


51 


5 


6857 


6908 


6959 


7011 


7062 


7114 


7165 


7216 


7268 


7319 


51 


6 


7370 


7422 


7473 


7524 


7576 


7627 


7678 


7730 


7781 


7832 


51 


7 


7883 


7935 


7986 


8037 


8088 


8140 


8191 1 8242 


8293 


8345 


51 


8 


8396 


8447 


8498 


8549 


8601 


8052 


87031 8754 


8805 


8857 


51 


9 


8908 


8959 


9010 


9061 


0112 


9163 


9215 


9266 


9317 


9368 


51 


850 


929419 


929470 


929521 


929572 


929623^929674 


929725 


929776 


929827 


929879 


51 


1 


9930 


9981 


930032 


930083 


930134 930185 


930236 


930287 


930338 


930389 


51 


2 


930440 


930491 


0542 


0592 


0643 j 0694 


0745 


0796 


0847 


0898 


51 


3 


0949 


1000 


1051 


1102 


1153 


1204 


1254 


1305 


1356 


1407 


51 


4 


145S 


1509 


1560 


1610 


1661 


1712 


1763 


1814 


1805 


1915 


51 


5 


1966 


2017 


2068 


2118 


2169 


2220 


2271 


2322 


2372 


2423 


51 


6 


2474 


2524 


2575 


2626 


2677 


2727 


2778 


2829 


2879 


2930 


51 


7 


2981 


3031 


3082 


3133 


3183 


3234 


3285 


3335 


3386 


3437 


51 


8 


3487 


3538 


3589 


3639 


3690 


3740 


3791 


3841 


3892 


3943 


51 


9 


3993 


4044 


4094 


4145 


4195 


4246 


4296 


4347 


4397 


4448 


51 


860 


934498 


934549 


934599 


934650 


934700 934751 


934801 


934852 


934902 


934953 


50 


1 


5003 


5054 


5104 


5154 


5205! 5255 


5306 


5356 


5406 


5457 


50 


2 


5507 


5558 


5608 


5658 


57091 5759 


5809 


5860 


5910 


5960 


50 


3 


6011 


0061 


6111 


6162 


62l2i 6262 


6313 


6363 


6413 


6463 


50 


4 


6514 


6564 


0614 


6665 


6715 6765 


6815 


6865 


6916 


6966 


50 


5 


7016 


7066 


7117 




7217 7267 


7317 


7367 


7418 


7468 


50 


6 


7518 


7568 


7618 


7668 


7718 7769 


7819 


7869 


7919 


7969 


50 


7 


8019 


8069 


8119 


8169 


8219! 8269 


8320 


8370 


8420 


8470 


50 


8 


8520 


8570 


8620 


8670 


87201 8770 


8820 


8870 


8920 


8970 


50 


9 


9020 


9070 


9120 


9170 


92201 9270 


9320 


9369 


9419 


9469 


50 


870 


939519 


939569 


939619 


939669 


939719*939769 


839819 


939869 


939918 


939968 


50 


1 


940018 


940068 


9401 18 


94016S 


940218 940267 


940317 


940307 


940417 


940467 


50 


2 


0516 


0566 


06J6 


0666 


0716 0765 


0815 


0865 


0915 


0964 


50 


3 


10141 1064 


1114 


1163 


1213 1263 


1313 


1362 


1412 


1462 


50 


4 


1511 


1561 


1611 


1660 


1710 1760 


1809 


1S59 


1909 


1958 


50 


5 


2008 


2058 


2107 


2157 


2207' 225() 


2306 


2355 


2405 


2455 


50 


6 


2504 


2554 


2603 


2653 


2702 2752 


280 1 


2851 


2901 


2950 


50 


7 


3000 


3049 


3009 


3148 


3198 3247 


3297 


3346 


3396 


3445 


49 


8 


3495 


3544 


3593 


3643 


3692 3742 


379! 


3841 


3890 


3939 


49 


9i 39d9 


4038 


4088 4137 


4186 4236 


4285 


4335 


4384 


4433 


49 



Np, I I I j J§ I 3 1 4 I 5 



i 7 I 8 



9 'D^ 



194 



LOGARITHMS OF NUMBERS. 



Na I O I 1 



I 3 



* I 



I 7 I 8 I 9 iDiff 



880 
1 
2 
3 
4 
5 
6 
7 
8 
9 

890 
1 
2 
3 
4 
5 
6 
7 
8 
9 

900 
1 
2 
3 
4 
5 



910 
1 
2 
3 
4 
5 
6 
7 
8 
9 

9*20 



930 



944483 


944532 


944581 


944631 


944680 


944729 


944779 


944828 


4976 


5025 


5074 


5124 


5173 


5222 


5272 


5321 


5469 


5518 


5567 


5616 


5665 


5715 


5764 


5813 


5961 


6010 


6059 


6108 


6157 


6207 


6256 


6305 


6452 


6501 


6551 


6600 


6649 


6698 


0747 


6796 


6943 


6992 


7041 


7090 


7140 


7189 


7238 


7287 


7434 


7483 


7532 


7581 


7630 


7679 


7728 


7777 


7924 


7973 


8022 


8070 


8119 


8168 


8217 


8266 


8413 


8462 


8511 


8560 


8609 


8657 


8706 


8755 


8902 


8951 


8999 


9048 


9097 


9146 


9195 


9244 


949390 


949439 


949488 


949536 


949585 


949634 


949083 


949731 


9878 


9926 


9975 


950024 


950073 


950121 


950170 


950219 


950365 


950414 


950462 


0511 


0560 


0608 


0657 


0706 


0851 


0900 


0941, 


0997 


1046 


1095 


1143 


1192 


1338 


1386 


1435 


1483 


1532 


1580 


1629 


1(577 


1823 


1872 


1920 


1969 


2017 


2066 


2114 


2163 


2308 


2356 


2405 


2453 


2502 


2550 


2599 


2647 


2792 


2841 


2889 


2938 


2986 


3(J34 


3083 


3131 


3276 


3325 


3373 


3421 


3470 


3518 


3566 


3615 


3760 


3808 


3856 


3905 


3953 


4001 


4049 


4098 


954243 


954291 


954339 


954387 


954435 


954484 


954532 


954580 


4725 


4773 


4821 


4869 


4918 


496(3 


5014 


5062 


5207 


5255 


5303 


5351 


5399 


5447 


5495 


5543 


5688 


5736 


5784 


5832 


5880 


5928 


5976 


6024 


6168 


6216 


6265 


6313 


6361 


6409 


6457 


6505 


6C49 


6697 


6745 


6793 


6840 


6888 


6936 


6984 


7128 


7176 


7224 


72-2 


7^2.y 


73oe 


7416 


7464 


7607 


7655 


7703 


7751 


7799 


7847 


7894 


7942 


8086 


8134 


8181 


8229 


8277 


8325 


8373 


8421 


8564 


8612 


8659 


8707 


8755 


8803 


8850 


8898 


959041 


959089 


959137 


959185 


959232 


95J280 


959328 


959375 


9518 


9566 


9614 


9661 


9-.rJ 


9757 


9<S04 


9852 


91)95 


900042 


96UU9U 


960138 


9:i0j85 


960233 


9602-^1 


9(i.);i28 


960471 


0518 


0566 


l>lii3 


ubuJ 


UWj 


0756 


U«(M 


0946 


0:1114 


1U41 


1089 


1136 


1184 


123 J 


1279 


1421 


1469 


1516 


1563 


Kill 


1658 


170() 


1753 


1895 


1943 


1990 


2038 


2085 


2132 


2180 


2227 


2369 


'2417 


2464 


2511 


2559 


2606 


2653 


2701 


2843 


289U 


2937 


2985 


3032 


3U79 


3126 


3174 


3316 


3363 


3410 


3457 


3504 


3552 


3599 


364() 


963788 


963835 


963882 


963929 


963977 


964024 


964071 


964118 


426U 


4307 


4354 


4401 


4448 


4495 


4542 


459.* 


4731 


4778 


4825 


4872 


4919 


49()6 


5013 


506) 


5202 


5249 


5296 


5343 


5390 


5437 


5484 


5531 


5672 


5719 


5766 


5813 


5860 


5907 


5954 


(iOOi 


6142 


6189 


6236 


6283 


6329 


6376 


6423 


(3470 


6611 


6658 


6705 


6752 


6799 


6845 


6892 


6939 


7080 


7127 


7173 


7220 


72; ,7 


73J4 


7361 


7408 


7548 


7595 


7642 


7688 


7735 


7782 


7H29 


7875 


8U1() 


8002 


8109 


8156 


8203 


8249 


8296 


8343 


968483 


968530 


968576 


968623 


968670 


968716 


968763 


968810 


8950 


8996 


9043 


9090 


9136 


9183 


9229 


9276 


9416 


9463 


9509 


955t) 


9()02 


9649 


9695 


9742 


9882 


9928 


9975 


970021 


970068 


970114 


970161 


970207 


970347 


970393 


970440 


048() 


05.J3 


'0579 


0(')2() 


0(572 


0812 


0858 


0904 


0951 


09ii7 


1044 


1090 


1137 


1276 


1322 


1369 


14J5 


1461 


1508 


1554 


1601 


1740 


1786 


1832 


1879 


1925 


1971 


2018 


20(;4 


2203 


2249 


2295 


2342 


2388 


2434 


2481 


2527 


2666 


2712 


2758 


28(M 


2851 


2897 


2943 


2989 



944877 1944927 



5370 

58(32 
6354 
6845 
7336 
7826 
8315 
8804 
9292 

949780 
950267 
0754 
1240 
1726 
2211 
2696 
3180 
3363 
4146 

954628 
5J10 
5592 
6072 
6553 
7032 
7512 
7990 
8468 
8946 

959423 
9900 

96i'376 
08." 1] 
1326 
1801 
2275 
2748 
3221 
3f);13 

964165 
4(i37 
5108 
5578 
6048 
6517 
6986 
7454 
7922 
8390 

968856 
9323 
9789 

970254 
0719 
1183 
1647 
2110 
2573 
3035 



5419 
5912 
6403 

6894 
7385 
7875 
8364 
8853 
9341 

949829 
950316 
0803 
1289 
1775 
2260 
2744 
3228 
3711 
4194 

954677 
5158 
5640 
6120 
6601 
7080 
7559 
8038 
8516 
8994 

959471 
9947 

960423 
0899 
1374 
1848 
2322 
2795 
3268 
3741 

964212 
4684 
5155 
5625 
6095 
()5(J4 
7033 
7501 
79()9 
843(3 

968903 
93(>9 
9835 

970300 
0765 
1229 

2157 
21) 19 
3082' 



|0|l|»i3)4:|d|617|8|9|Oiff 



LOGARITHMS OF XUJIBERS. 



I 3 I 3 I 4 I 5 



6 19-1 



I 9 iDtfl 



940 


973128 


973174 


973220 


973266 


973313 


973359 


973405 


97345! 


973497 


973543 


46 


1 


3590 


3636 


3()82 


3728 


3774 


3820 


3866 


3913 


3959 


4005 


46 


2 


4051 


4097 


4143 


4189 


4235 


4281 


4327 


4374 


4420 


4466 


46 


3 


' 4512 


4558 


4004 


4650 


4696 


4742 


4788 


4834 


4880 


4926 


46 


4 


4972 


5018 


5(J04 


5110 


515e 


5202 


5248 


5294 


5340 


5386 


46 


5 


543i> 


5478 


5524 


5570 


5616 


5662 


5707 


5753 


5799 


5845 


46 


6 


5891 


5937 


5983 


0029 


6075 


6121 


0107 


6212 


6258 


()304 


46 


7 


6350 


6396 


6442 


6488 


6533 


6579 


6625 


6671 


0717 


6763 


46 


8 


6808 


6H54 


6900 


6946 


6992 


7037 


7083 


7129 


7175 


7220 


40 


y 


7260 


7312 


7358 


7403 


7449 


7495 


7541 


7586 


7032 


7678 


46 


gsu 


977724 


977769 


977815 


977801 


977900 


977952 


977998 


978043 


978089 


978 J 35 


46 


1 


8181 


8226 


8272 


8317 


8363 


8409 


8454 


8500 


8540 


8591 


46 


2 


8637 


8683 


8728 


8774 


8819 


88()5 


8911 


8J50 


9002 


9047 


46 


3 


9093 


9138 


9184 


9230 


9275 


9321 


9366 


9412 


9457 


9503 


46 


4 


9548 


9594 


9039 


9085 


9730 


9770 


9821 


9867 


9912 


9958 


46 


5 


980003 


980049 


980094 


980140 


980185 


980231 


980270 


980322 


980307 


980412 


45 


fi 


0458 


0503 


0549 


0594 


(JG40 


0085 


0730 


0770 


0821 


0867 


45 


7 


0912 


0957 


1003 


1048 


1093 


1139 


1184 


1229 


1275 


1320 


45 


8 


1300 


1411 


1450 


1501 


1547 


1592 


1637 


I6s:i 


1728 


1773 


45 


9 


1819 


1804 


1909 


1954 


2000 


2045 


2090 


2135 


2181 


2226 


45 


960 


982271 


982310 


982302 


982407 


982452 


982497 


982543 


982588 


982()33 


982678 


45 


1 


2723 


2709 


2814 


2859 


2904 


2949 


29!:4i 3040 


3085 


3130 


45 


2 


3175 


3220 


3205 


3310 


3350 


3401 


34;0| 3491 


3536 


3581 


45 


3 


3{>20 


3(i71 


3716 


3702 


3807 


3852 


38j71 3942 


2M1 


4032 


45 


4 4077 


4122 


4107 


4212 


4257 


4302 


4347 4392 


4437 


4482 


45 


5 4527 


4572 


4017 


4002 


4707 


4752 


4797 4842 


4887 


4932 


45 


G 4977 


5022 


5007 


5112 


5157 


5202 


5247 5292 


5337 


5382 


45 


7 5420 


5471 


5510 


5501 


5000 


5051 


5090 


5741 


5780 


5830 


45 


8 5875 


592(» 


5905 


0010 


0055 


6100 


6144 


6189 


6234 


6279 


45 


9| 0324 


6:509 


0413 


6458 


6503 


6548 


0593 


6637 


6682 


6727 


45 


970 980772 


980817 


980801 


986906 


986951 


986996 


987040 


987085 


987130 


987175 


45 


1 7219 


7204 


7309 


7353 


7398 


7443 


7488 


7532 


7577 


7622 


45 


2 7000 


7711 


7750 


7800 


7845 


7890 


7934 


7979 


8024 


8068 


45 


3 8113 


8157 


8202 


8247 


8291 


8336 


8381 


8425 


8470 


8514 


45 


41 8559 


8G04 


8048 


8693 


8737 


8782 


8826 


8871 


8910 


8960 


45 


5| 9005 


9049 


9094 


9138 


9183 


9227 


9272 


9310 


9361 


9405 


45 


945U 


9494 


9539 


9583 


9028 


9672 


9717 


9701 


9806 


9850 


44 


7 9895 


9939 


9983 


990028 


990072 


990117 


990161 


990200 


990250 


990294 


44 


8 990339 


990;i83 


990428 


0472 


0510 


0501 


0005! OboO 


0094 


0738 


44 


9 0783 


0c^27 


0871 


0916 


0900 


1004 


1049 


1093 


1137 


1182 


44 


980 99122G 


991270 


991315 


991359 


991403 


991448 


991492 


991536 


991580 


991625 


44 


1 1009 


IT 13 


1758 


1802 


1840 


189U 


1935 


1979 


2023 


2067 


44 


% 2111 


2J50 


2200 


2244 


2288 


2333 


2377 


2421 


2465 


2509 


44 


3| 2554 


2598 


2042 


2086 


2730 


2774 


2819 


2803 


2907 


2951 


44 


4 '2995 


3(J39 


3083 


3127 


3172 


3210 


3300 


3304 


3348 


3392 


44 


5 3436 


3480 


3524 


3568 


3613 


3657 


3701 


3745 


3789 


383:i 


44 


6 3877 


3921 


3965 


4009 


4053 


4097 


414J 


4185 


4229 


4273 


44 


7 4317 


4301 


4405 


4449 


4493 


4537 


4581 


4025 


4009 


4713 


44 


8 4757 


48U1 


4845 


4889 


4933 


4977 


5021 


5005 


5108 


5152 


44 


9j 5190 


5240 


52t4 


5328 


5372 


5416 


5400 


5504 


5547 


5591 


44 


990 995635 


995(;79 


995723 


995767 


995811 


(,95854 


995898 


995942 


995980 


990030 


44 


1 


6074 


0J17 


0101 


6205 


0249 


6293 


0337 


0380 


6424 


041 KM 


44 


2 


6512 


0555 


6599 


6643 


6087 


6731 


0774 


0818 


0862 


09U(i 


44 


3 


6949 


6993 


7037 


7080 


7124 


7108 


72 J 2 


7255 


7299 


7343 


44 


4 


7386 


7430 


7474 


7517 


7561 


7005 


7048 1 7692 


7736 


7779 


44 


5 


7823 


7807 


7910 


7954 


7998 


8041 


8085 


8129 


8i72 


8210 


44 


6 


8259 


8303 


8347 


8390 


8434 


8477 


8521 


8564 


8608 


805-_> 


44 


7 


8695 


8739 


8:82 


8820 


8869 


8913 


8950 


9000 


9043 


9087 


44 


8 


9131 


9174 


9218 


920J 


9305 


934H 


9392 


9435 


9479 


9522 


44 


_9 


9565 


9009 


9652 


9696 


9739 


9783 


9826 


9870 


9913 


9957 


4J 



No. I O I 1 I a I 3 



5 I 6 j ? I 8 » 9 IDjS 



TABLE XIII. 

LOGARITHMIC SINES, C@SINES, TANGENTS, AND 
COTANGENTS. 



N. B. — The minutes in the left-hand column of each page, 
increasing downwards, belong to the degrees at the top ; and 
those increasing upwards, in the right-hand column, helone to 
the degrees below. 

In using the differences for one second, in columns D, the 
two right-hand figures should be marked off as decimals. 
Thus the difference for log, sin. 1° 12' 5" would be 99.83 
X 5 = 499.1, additive to the mantissa .321027 treated as an 
integer, and the difference for log. cos. 8° 30' 50" would be 
0.30 X 50 = 16.0 subtractive from the mantissa .995203 treated 
us an integer. 

The differences in columns D range opposite the upper one 
of the two functions to which they respectively apply. 

The first column D refers to Sines, the second to Cosines, 
the third to both Tangents and Cotangents. 



V>' 



ICS (0]ic,LcrcfO LOGAEITHMIC ZJNES, COSIXKS. ETC. 



M. 


Sine 


D. 


Cosine 


! D. 


1 Tan?. 

0-000000 


D. 


Cotansr. 

Inttniie. 


60 


(1 


Inf. Ne;,'. 




1 0-000000 


I 


(i-4«>3726 


501717 


000000 


00 


6-463726 


501717 


13-536274 


59 


2 


704756 


293485 


000000 


00 


764756 


293485 


235^44 


58 


:i 


940847 


208231 


000000 


00 


940847 


208231 


059153 


.57 


4 


7-()65786 


J615I7 


000000 


00 


7-0(55786 


161517 


12-934214 


.56 


,") 


162696 


131968 


000000 


00 


Iti2696 


131969 


837304 


55 


() 


241877 


1 1 1578 


9-999999 


01 


241878 


111578 


758122 


54 


7 


308824 


96(i53 


999999 


01 


308825 


99653 


691175 


.53 


8 


366816 


85254 


999999 


01 


366817 


85254 


&33I83 


52 


9 


417968 


76263 


999i)99 


01 


417970 


762(53 


582030 


51 


10 


463725 


68988 


999998 


01 


463727 


68988 


536273 


.50 > 


11 


7505] 18 


62981 


9-999998 


01 


7-505120 


6298; 


i2-494H80 


49. ' 


12 


542906 


57936 


999997 


01 


542909 


57938 


457091 


48 


13 


577668 


53641 


999997 


01 


577672 


53642 


4-2-2:i->8 


47 


14 


609853 


49938 


999996 


01 


609857 


49939 


390143 


46 


15 


639816 


A^' !4 


999996 


01 


639820 


46715 


3(;()180 


45 


If) 


667845 


-)3H8J 


999995 


01 


6(57819 


43882 


332151 


44 


17 


694173 


41372 


999995 


01 


694179 


41373 


305821 


43 


18 


718997 


39135 


999994 


0! 


7!9(M.'3 


39136 


280it97 


42 


19 


742477 


37127 


999993 


01, 


742484 


37128 


257516 


41 


'^0 


7(M754 


35315 


999993 


01 


764761 


35317 


235239 


40 


21 


7 785943 


33672 


9-999992 


01 


7-785951 


33673 


12-214049 


39 


22 


806146 


32175 


999991 


01 


806155 


32176 


193845 


38 


23 


825451 


30805 


999990 


01 


825460 


30806 


174540 


37 


24 


843934 


29547 


9999P9 


02 


843944 


29549 


156056 


36 


25 


861662 


28388 


999988 


02 


861674 


28390 


138326 


35 


2() 


878695 


27317 


999988 


02 


878708 


27318 


121292 


.34 


'}? 


895085 


2()323 


999987 


(12 


895099 


26325 


104901 


33 


2t; 


910879 


25399 


999986 


02 


910894 


25401 


0891(i6 


32 


29 


926J19 


24538 


999985 


02 


926 Kll 


24540 


0738(5(5 


31 


30 


940842 


23733 


999983 


02 


940858 


23735 


059142 


30 


31 


7-955082 


22980 


9-999982 


02 


7-955100 


22981 


12044900 


29 


32 


96H87() 


22273 


9999!^ 1 


02 


968889 


22275 


031111 


28 


33 


982233 


21(108 


999980 


02 


982253 


21610 


017747 


27 


34 


995198 


2ilit8l 


999979 


02 


995219 


20983 


004781 


26 


35 


8-U07787 


20390 


999977 


02 


8-',|(l78(19 


2(1392 


11 ■992191 


25 


3C. 


020021 


19H3i 


999!t76 


02 


(12(1045 


19833 


979955 


24 


37 


(>3»'.)I9 


19302 


999975 


02 


o;U945 


19305 


9(58055 


23 


3H 


043501 


I8r'0l 


999973 


02 


043527 


18803 


956473 


22 


39 


05478 1 


18:^25 


999972 


02 


054809 


18327 


945191 


21 


40 


065776 


17872 


999971 


02 


065806 


17874 


934194 


20 


41 


8076500 


17441 


9-999969 


02 


8-07(553 1 


17444 


11-923469 


19 


42 


086965 


17031 


999968 


02 


086997 


17034 


913003 


18 


43 


097183 


16639 


999966 


02 


097217 


1(5(542 


902783 


17 


44 


1(17167 


16265 


999964 


03 


107202 


162(58 


892797 


?6 


45 


116926 


J 5908 


999963 


03 


1 1(5963 


15910 


883037 


J5 


4G 


126471 


15566 


999961 


03 


12(5510 


15568 


873490 


14 


47 


135HI0 


15238 


999959 


0.-. 


135H5I 


15241 


8(54149 


13 


48 


144953 


14924 


999958 


(.3 


144996 


14927 


855004 


12 


49 


15391)7 


1-1622 


999956 


03 


153952 


14627 


846048 


11 


50 


1626.'J| 


1433;; 


999954 


03 


162727 . 


14336 


837273 


10 


51 


8l7i^2H() 


14054 


9-999952 


03 


8- 171328 


14057 


11-828(572 


9 


52 


179713 


13786 


999950 


03 


179763 


13790 


820237 


8 


53 


187985 


13529 


999948 


03 


188036 


13532 


811964 


7 


54 


I'M;|fi2 


13280 


999946 


03 


196156 


13284 


803844 


6 


55 


2iM07(» 


13041 


999944 


03 


204126 


13044 


795874 


5 


5() 


211895 


12810 


999942 


04 


211953 


12?<I4 


788047 


4 


57 


219581 


12.587 


999940 


04 


219641 


12590 


780359 


3 


58 


227134 


12372 


999938 


04 


227195 


12376 


772805 


2 


59 


234557 


12164 


999936 


04 


234621 


12168 


765379 


1 


60 


241855 


11963 


999934 


J)4_ 


241921 


11967 


758079 







1 Co««e 




1 Siae 


I 


Cotane- 1 


1 


Tang. 1 








891 


[)egr« 


«S. 











LOGARITHMIC SmES 


COSIh^ES, ETC. (1 D 


egrcc.) 


\m 


si. 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 1 


Cotan?. 







8-241855 


11963 


9-999934 


04 


8-241921 


11967 


11-758079 


'm 


1 


249033 


11768 


999932 


04 


249102 


11772 


. 750898 


59 


2 


256094 


11580 


999929 


04 


256165 


11584 


743835 


58 


3 


263042 


11398 


999927 


14 


263115 


11402 


736885 


57 


4 


269881 


11221 


999925 


04 


269956 


11225 


730044 


56 


5 


276614 


11050 


999922 


04 


276691 


11054 


723309 


55 


6 


283243 


10883 


999920 


04 


283323 


10887 


716677 


54 


7 


289773 


10721 


999918 


04 


289856 


10726 


710144 


53 


8 


296207 


10565 


999915 


04 


296292 


10570 


703708 


52 


9 


302546 , 


10413 


999913 


04 


302634 


10418 


697366 


51 


10 


308794 


10266 


999910 


04 


308884 


10270 


691116 


50 


11 


8-314954 


10122 


9-999907 


04 


8-315046 


10126 


11-684954 


49 


12 


321027 


9982 


999905 


04 


321122 


9987 


678878 


48 


J3 


327016 


9847 


999902 


04 


327114 


9851 


672886 


47 


14 


332924 


9714 


999899 


05 


333025 


9719 


666975 


46 


15 


338753 


9586 


999897 


05 


338856 


9590 


661144 


45 


If) 


344504 


9460 


999894 


05 


344610 


94r)5 


655390 


44 


J7 


350181 


9338 


999891 


05 


350289 


9343 


649711 


43 


18 


355783 


9219 


999888 


05 


355895 


9224 


644105 


42 


19 


361315 


9103 


999885 


05 


361430 


9108 


638570 


41 


20 


366777 


8990 


999882 


05 


366895 


8995 


633105 


40 


21 


8-372171 


8880 


9-999879 


05 


8-372292 


8885 


11-627708 


39 


22 


377499 


8772 


999876 


05 


377622 


8777 


622378 


38 


23 


382762 


86t)7 


999873 


05 


382889 


8672 


617111 


37 


24 


387962 


8564 


999870 


05 


388092 


8570 


611908 


36 


25 


393101 


8464 


999867 


05 


393234 


8470 


606766 


35 


26 


398179 


8366 


999864 


05 


398315 


8371 


601685 


34 


27 


403199 


8271 


999861 


05 


403338 


8276 


596662 


33 


28 


408161 


8177 


999858 


05 


408304 


8182 


591696 


32 


29 


413068 


8086 


999854 


05 


413213 


8091 


586787 


31 


30 


417919 


7996 


999851 


06 


418068 


8002 


581932 


30 


31 


8-422717 


7909 


9-999848 


06 


8-422869 


7914 


11-577131 


29 


32 


427462 


7823 


999844 


06 


427618 


7830 


572382 


28 


33 


432156 


7740 


999841 


06 


432315 


7745 


567685 


27 


34 


43G800 


7657 


999838 


06 


436962 


7663 


563038 


26 


35 


441394 


7577 


999834 


06 


441560 


7583 


558440 


25 


36 


445941 


7499 


999831 


06 


446110 


7505 


553890 


24 


37 


450440 


7422 


999827 


06 


450613 


7428 


549387 


23 


38 


454893 


7346 


999823 


06 


455070 


7352 


544930 


22 


39 


459301 


7273 


999820 


06 


459481 


7279 


540519 


21 


40 


463665 


7200 


999816 


06 


463849 


7206 


536151 


20 


41 


8-467985 


7129 


9-999812 


06 


8-468172 


7135 


11-531828 


19 


42 


472263 


7060 


999809 


06 


472454 


7066 


527546 


18 


43 


47C498 


6991 


999805 


06 


476693 


6998 


523307 


17 


44 


480693 


6924 


999801 


06 


480892 


6931 


519108 


16 


45 


484848 


6859 


999797 


07 


485050 


6865 


514950 


15 


46 


488963 


6794 


999793 


07 


489170 


6801 


510830 


14 


47 


493040 


6731 


999790 


07 


493250 


6738 


506750 


13 


48 


497078 


6669 


999786 


07 


497293 


6676 


5112707 


12 


49 


501080 


6608 


999782 


07 


501298 


6615 


498702 


11 


50 


505045 


6548 


999778 


07 


505267 


6555 


494733 


10 


51 


8-508974 


6489 


9-999774 


07 


8-509200 


6496 


11-490800 


9 


52 


512867 


6431 


999769 


07 


513098 


6439 


486902 


8 


53 


51(i726 


6375 


999765 


07 


516961 


6382 


483039 


7 


54 


520551 


6319 


999761 


07 


520790 


6326 


479210 


6 


55 


524343 


6264 


999757 


07 


524586 


6272 


475414 


5 


56 


5-28102 


6211 


989753 


07 


528349 


6218 


471651 


4 


57 


531828 


6158 


999748 


07 


532080 


6165 


467920 


3 


58 


535523 


6106 


999744 


07 


535779 


6113 


464221 


2 


59 


539186 


6055 


999740 


07 


539447 


6062 


460553 


1 


60 


542819 


6004 


999735 


07 


543084 


6012 


456916 







1 Coaioc 1 


J 


Sine 
88 


Degr© 


Cotang. 
e& 


i 


1 Tauff. 


11 



200 (2 Degrees.) LOGARITHMIC SINES, CO SIXES, ETC. 



M. 


Sine 


D. 


Cosine 


D. 


Tariff. 


D. 


Cotang-. 







8-542819 


6004 


9-999735 


07 


8-543084 


6012 


11456916 


60 


1 


546422 


5955 


999731 


07 


546691 


5962 


453309 


59 


2 


549995 


5906 


999726 


07 


550268 


5914 


449732 


58 


3 


553539 


5858 


999722 


08 


553817 


5866 


44(5183 


57 


4 


557054 


5811 


999717 


08 


557336 


5819 


442664 


56 


5 


560540 


5765 


999713 


08 


56082S 


5773 


439172 


55 


6 


563999 


5719 


999708 


08 


564291 


5727 


435709 


54 


7 


567431 


5674 


990704 


08 


567727 


5682 


432273 


53 


8 


570836 


5630 


999699 


08 


571137 


5638 


428863 


52 


9 


574214 


5587 


999694 


08 


574520 


5595 


425480 


51 


JO 


577566 


5544 


999689 , 


08 


577877 


5552 


422123 


50 


Jl 


8-580892 


5SU2 


9-999685 


08 


8-581208 


5510 


11-418792 


49 


12 


584193 


5460 


999680 


08 


584514 


5468 


415486 


48 


13 


587469 


5419 


999675 


08 


587795 


5427 


412205 


47 


14 


590721 


5379 


999670 


08 


591051 


5387 


408949 


46 


15 


593948 


5339 


999665 


08 


594283 


5347 


405717 


45 


IG 


597152 


5300 


999660 


08 


597492 


5308 


402508 


44 


17 


600332 


5261 


999655 


08 


600677 


5270 


399323 


43 


18 


603489 


5223 


999650 


08 


603839 


5232 


396161 


42 


19 


606623 


5186 


999645 


09 


606978 


5194 


393022 


41 


20 


609734 


5149 


999640 


09 


610094 


5158 


389906 


40 


21 


8-612823 


5112 


9-999635 


09 


8-613189 


5121 


11-386811 


39 


22 


615891 


5076 


999629 


09 


616262 


5085 


383738 


38 


23 


618937 


5041 


999624 


09 


619313 


5050 


380687 


37 


24 


621962 


5006 


999619 


09 


622343 


5015 


377657 


36 


25 


624965 


4972 


999614 


09 


625352 


4981 


374648 


35 


26 


627948 


4938 


999608 


09 


628340 


4947 


371660 


34 


27 


630911 


4904 


999603 


09 


631308 


4913 


368692 


33 


28 


633854 


4871 


999597 


09 


634256 


4880 


365744 


32 


29 


636776 


4839 


999592 


09 


637184 


4848 


362816 


31 


30 


639680 


4806 


999586 


09 


640093 


4816 


359907 


30 


31 


8642563 


4775 


9-999581 


09 


8-642982 


4784 


11-357018 


29 


32 


645428 


4743 


999575 


09 


645853 


4753 


354147 


28 


33 


648274 


4712 


999570 


09 


648704 


4722 


351296 


27 


34 


651102 


4682 


999564 


09 


651537 


4691 


348463 


26 


35 


653911 


4652 


999558 


10 


654352 


4661 


345648 


25 


36 


656702 


4622 


999553 


10 


657149 


4631 


342851 


24 


37 


659475 


4592 


999547 


10 


659928 


4602 


340072 


23 


38 


662230 


4563 


999541 


10 


662689 


4573 


337311 


22 


39 


664968 


4535 


999535 


10 


665433 


4544 


334507 


21 


40 


667689 


4506 


999529 


10 


668160 


4526 


331840 


20 


41 


8670393 


4479 


9-999524 


10 


8-670870 


4488 


11-329130 


19 


42 


673080 


4451 


999518 


10 


673563 


4461 


326437 


18 


43 


675751 


4424 


999512 


10 


676239 


4434 


323761 


17 


44 


678405 


4397 


999506 


10 


678900 


4417 


321100 


16 


45 


681043 


4370 


999500 


10 


681544 


4380 


318456 


15 


46 


683665 


4344 


999493 


10 


684172 


4354 


315828 


14 


47 


686272 


4318 


999487 


10 


686784 


4328 


313216 


13 


48 


688863 


4292 


999481 


10 


689381 


4303 


310619 


12 


49 


691438 


4267 


999475 


10 


691963 


4277 


308037 


11 


50 


693998 


4242 


999469 


10 


694529 


4252 


305471 


10 


51 


8-696543 


4217 


9-999463 


11 


8-697081 


4228 


11-302919 


9 


52 


699073 


4192 


999456 


11 


699617 


4203 


300383 


8 


5.$ 


701589 


4168 


999450 


11 


702139 


4179 


297861 


7 


54 


704090 


4144 


999443 


11 


704646 


4155 


295354 


6 


55 


706577 


4121 


999437 


11 


707140 


4132 


292860 


5 


56 


709049 


4097 


999431 


11 


709618 


4108 


290382 


4 


57 


711507 


4074 


999424 


11 


712083 


4085 


287917 


3 


58 


713952 


4051 


999418 


11 


714534 


4062 


285465 


2 


59 


716383 


4029 


999411 


11 


716972 


4040 


283028 


I 


60 


718800 


4006 


999404 


11 


719396 


4017 


280604 






Sine I 



t Cotang. I 



I Tan-. 1 M. 



&IDVBK^ 



LOGARITHMIC SI^ES, COSINES, ETC. (3 Degrees.) 20\ 



M 


Sme 


D. 


Cosine 


D. I 


Tang. 1 


D. 1 


Cotnng. 







8-718800 


4006 


9-999404 


11 


8-719396 


4017 


11-280604 


60 


1 


721204 


3984 


999398 


11 


721806 


3995 


278194 


59 


2 


723595 


3962 


999391 


11 


724204 


3974 


275796 


58 


3 


725972 


3941 


999384 


11 


726588 


3952 


273412 


57 


4 


728337 


3919 


999378 


11 


728959 


3930 


271041 


56 


5 


730688 


3898 


999371 


11 


731317 


3909 


268683 


55 


6 


733027 


3877 


999364 


12 


733663 


3889 


266337 


54 


7 


735354 


3857 


999357 


12 


735996 


3868 


264004 


53 


8 


737067 


3836 


999350 


12 


738317 


3848 


261683 


52 


9 


7399G9 


3816 


999343 


12 


740626 


3827 


259374 


51 


10 


742259 


3796 


999336 


12 


742922 


3807 


257078 


50 


]1 


8-744536 


3776 


9-999329 


12 


8-745207 


3787 


11.254793 


49 


12 


740802 


3756 


999322 


12 


747479 


3768 


252521 


48 


13 


749055 


3737 


999315 


12 


749740 


3749 


25C260 


47 


14 


751297 


3717 


999308 


12 


751989 


3729 


24801 : 


46 


15 


753528 


3698 


999301 


12 


754227 


3710 


245773 


45 


16 


755747 


3679 


999294 


12 


750453 


3092 


243547 


44 


17 


757955 


3661 


999286 


12 


758668 


3673 


241332 


43 


18 


700151 


3642 


999279 


12 


760872 


3655 


239128 


42 


19 


762337 


3624 


999272 


12 


763065 


3636 


230935 


41 


20 


704511 


3606 


999265 


12 


765246 


3618 


234754 


40 


21 


8-766675 


3588 


9-999257 


12 


8-7U7417 


3600 


11-232583 


39 


22 


768828 


3570 


999250 


13 


769578 


3583 


230422 


38 


23 


770970 


3553 


999242 


13 


771727 


3565 


228273 


37 


24 


773101 


3535 


999235 


13 


773806 


3548 


226134 


36 


25 


775223 


3518 


999227 


13 


775995 


3531 


224005 


35 


26 


777333 


3501 


999220 


13 


778114 


3514 


22188G 


34 


27 


779434 


3484 


999212 


13 


780222 


3497 


219778 


33 


28 


781524 


3407 


993205 


13 


782320 


3480 


217680 


32 


29 


783005 


3451 


999197 


13 


784408 


3464 


215592 


31 


30 


785075 


3431 


999189 


13 


786486 


3447 


213514 


30 


31 


8-787736 


3418 


9-999181 


13 


8-788554 


3431 


11-211446 


29 


32 


789787 


3402 


999174 


13 


790613 


Ml 4 


209387 


28 


33 


791828 


3386 


999166 


13 


792G62 


3399 


207338 


27 


34 


793859 


3370 


999158 


13 


794701 


3383 


205299 


26 


35 


795881 


3354 


999150 


13 


796731 


3368 


203269 


25 


36 


797894 


3339 


999142 


13 


798752 


3352 


201248 


24 


37 


799897 


3323 


999134 


13 


800763 


3337 


199237 


23 


38 


801892 


3308 


999126 


13 


802765 


3322 


197235 


22 


39 


803876 


3293 


999118 


13 


804758 


3307 


195242 


21 


40 


805852 


3278 


999110 


13 


806742 


3292 


193258 


20 


41 


8-807819 


3263 


D-999102 


13 


8-808717 


3278 


11191283 


19 


42 


809777 


3249 


999094 


14 


810683 


32G2 


189317 


18 


43 


811726 


3234 


993080 


14 


812641 


3248 


187359 


17 


44 


813G67 


3219 


999077 


14 


814589 


3233 


185411 


16 


45 


815599 


3205 


999009 


14 


816529 


5219 


183471 


15 


46 


817522 


3191 


993061 


14 


818461 


3205 


181539 


14 


47 


819436 


3177 


999053 


14 


820384 


3191 


179616 


13 


48 


821343 


3103 


999044 


14 


822298 


3177 


177702 


12 


49 


823240 


3149 


999036 


14 


824205 


3103 


175795 


11 


50 


825130 


3135 


999027 


14 


826103 


3150 


173897 


10 


51 


8-827011 


3122 


9-999019 


14 


8-827992 


3130 


11-172008 


9 


52 


828884 


3108 


999010 


14 


829874 


3123 


170126 


8 


53 


830749 


3095 


999002 


14 


831748 


3110 


168252 


7 


54 


832G07 


3082 


998993 


14 


833613 


3096 


166387 


6 


55 


834456 


3009 


998984 


14 


835471 


3083 


164529 


5 


56 


836297 


305G 


998976 


14 


837321 


3070 


162679 


4 


57 


838130 


3043 


998967 


15 


839163 


3057 


160837 


3 


58 


839956 


3030 


998958 


15 


840998 


3045 


159002 


2 


59 


84171i 


3017 


998950 


15 


842825 


3032 


157175 


1 


60 


843585 


3000 


998941 


15 


844644 


3019 


155356 






I Cosme 



Sine ,1 I Cotang. 

86 P agrees, 



I -pang. I M. 



202 (4 Degrees.) LOGARITH^TIG SINES, CVSINES, ETC. 



M. I Sine 



I Cosine I U. 1 Tanp:- 1 



1 Cotang. I 






8-843585 


3005 


9-998941 


15 


8-844644 


3019 


11-155356 


60 


1 


845387 


2992 


998932 


15 


846455 


3007 


153545 


59 


2 


847183 


2980 


998923 


15 


848260 


2995 


151740 


58 


3 


848971 


2967 


998914 


15 


850057 


2982 


149943 


57 


4 


850751 


2955 


998905 


15 


85 1 846 


2970 


148154 


56 


5 


852525 


2;»43 


998896 


15 


853028 


2:;58 


1-IG372 


55 


6 


854291 


2931 


998887 


15 


855403 


z.m 


144597 


54 


7 


85G049 


29J9 


998878 


15 


857171 


2935 


142829 


53 


8 


857801 


2907 


99S8G9 


15 


858932 


2923 


141068 


52 


9 


859546 


2896 


998800 


15 


833G8G 


2911 


139314 


51 


10 


861283 


2884 


998851 


15 


802433 


2.X0 


137537 


50 


11 


8-863014 


2873 


9-998841 


15 


8-834173 


2888 


11 1358-37 


49 


]2 


884738 


2861 


998832 


15 


805906 


2877 


134094 


48 


13 


866455 


2850 


998823 


16 


837G32 


233G 


132338 


47 


14 


868165 


2839 


998813 


16 


869351 


2854 


130649 


46 


15 


869868 


2828 


998804 


16 


871004 


2843 


128936 


45 


16 


871565 


2817 


998795 


16 


872770 


2832 


127230 


44 


17 


873255 


2806 


998785 


16 


874469 


2821 


125531 


43 


18 


874938 


2795 


998776 


16 


876162 


2811 


123838 


42 


19 


876615 • 


2786 


9987G6 


16 


877849 


2800 


122151 


41 


20 


878285 


2773 


998757 


16 


879529 


2789 


120471 


40 


21 


8-879949 


2763 


9-998747 


16 


8-881202 


2779 


11-118798 


39 


22 


881607 


2752 


998738 


16 


882839 


2708 


117131 


38 


23 


883258 


2742 


998728 


16 


884530 


2758 


115470 


37 


24 


884903 


2731 


998718 


ICi 


883185 


2747 


113815 


33 


25 


886542 


"721 


9987(;8 


IG 


8378;:3 


2737 


112 137 


35 


2G 


888174 


2711 


998699 


10 


889476 


2727 


110524 


34 


27 


889801 


2700 


998689 


IG 


89J112 


2717 


108888 


33 


28 


891421 


2G90 


998079 


IG 


892742 


2707 


107258 


32 


29 


893035 


2680 


998669 


17 


894366 


2697 


1C5G34 


31 


30 


894043 


2670 


998659 


17 


895984 


3387 


10401G 


39 


31 


8-896246 


2660 


9-998649 


17 


8-89759G 


2677 


11-102404 


29 


32 


897842 


2051 


998639 


17 


8992C3 


2367 


100797 


28 


33 


899432 


2641 


998629 


17 


900803 


2658 


099197 


07 


34 


901017 


2631 


998619 


17 


902398 


2C48 


097632 


23 


35 


902596 


2622 


998609 


17 


903987 


2638 


09G0I3 


25 


3() 


904169 


2612 


998599 


17 


905570 


2329 


094439 


24 


37 


905736 


2303 


998589 


17 


907147 


2620 


092853 


23 


38 


907297 


2593 


998578 


17 


908719 


2610 


091281 


22 


39 


908853 


2584 


998568 


17 


910285 


2601 


089715 


21 


40 


910404 


2575 


998558 


17 


911840 


2592 


088154 


20 


41 


8-911949 


2506 


9-998548 


17 


8-913401 


2583 


11-086599 


19 


42 


913488 


2556 


998537 


17 


914951 


2574 


085049 


18 


43 


915022 


2547 


998527 


17 


91C495 


2565 


083505 


17 


44 


916550 


2538 


9985 IC 


1Q 


918034 


255G 


0819C6 


10 


45 


9 J 8073 


2529 


998500 


18 


91 9508 


2547 


080432 


15 


46 


919591 


2520 


998495 


18 


921096 


2538 


078904 


14 


47 


921103 


2512 


998485 


18 


922619 


2530 


077381 


13 


48 


922610 


2503 


998474 


18 


924136 


2521 


075834 


12 


49 


924112 


2494 


998464 


18 


925()49 


2512 


074351 


11 


50 


925609 


2486 


998453 


18 


927 J 53 


2503 


072844 


10 


51 


8-927100 


2477 


9-998442 


18 


8-l!28(i58 


2495 


11-071342 


9 


52 


928587 


2469 


998431 


18 


'.)30155 


2486 


0G9845 


8 


53 


930068 


2460 


998421 


18 


931647 


2478 


068353 


7 


54 


931544 


2452 


998410 


18 


933134 


2470 


0G6866 


6 


55 


933015 


2443 


998399 


18 


934016 


2461 


065384 


5 


56 


934481 


2435 


998388 


18 


936093 


2453 


063907 


4 


57 


935942 


2427 


998377 


18 


937565 


2445 


062435 


3 


58 


937398 


2419 


998366 


18 


939032 


2437 


0609G8 


2 


59 


938850 


2411 


998355 


18 


940494 


2430 


059506 


1 


60 


940296 


2403 


998344 


18 


941952 


2421 


058048 






I Cosine I 



I 1 

85 Degrees 



Cotang. I 



Tang. I M, 



l^OGARITHMIC SINES, COSTXES, ETC. (S Dogrees.) 201: 



M. 


Sine 


D. 


Cosine 


1 D. 


Tang. 


1 u. 


Cotang. 




"o 


8-940296 


2403 


9-998344 


19 


8-941952 


2421 


11058048 


60 


1 


941738 


2394 


998333 


19 


943404 


2413 


05(55; M5 


59 


2 


943174 


2387 


998322 


19 


944f<52 


2405 


0.-)5148 


58 


3 


944606 


2379 


998311 


19 


946-J95 


2397 


053705 


57 


4 


946034 


2371 


998300 


19 


947734 


2390 


05-2266 


56 


5 


947456 


2363 


998289 


19 


949168 


2382 


050832 


55 


6 


94H874 


2355 


998277 


19 


950597 


2374 


049403 


54 


7 


950287 


2348 


998266 


19 


95-2021 


2366 


047979 


53 


8 


951696 


2340 


998255 


19 


953441 


2360 


046559 


52 


9 


9.>3I0(( 


2332 


998243 


19 


954^^5(5 


2351 


045144 


51 


in 


954499 


2325 


998232 


19 


956267 


2344 


043733 


53 


11 


8-95")894 


2317 


9-998220 


19 


8-957674 


2337 


11042326 


49 


12 


957284 


2310 


998209 


19 


959075 


2;}29 


040925 


48 


i:j 


958(570 


2302 


998197 


19 


960473 


2323 


039527 


47 


14 


9601152 


2295 


998186 


19 


961 S66 


2314 


038 K54 


46 


15 


961429 


2288 


998174 


19 


963255 


2307 


036745 


45 


Ifi 


9(>2-'i)l 


2280 


998163 


19 


9(54639 


2300 


0353(51 


44 


17 


964170 


2273 


998151 


19 


966019 


2293 


033981 


43 


18 


9(55534 


2266 


998139 


20 


967394 


2286 


032606 


42 


Vi 


906893 


2259 


998128 


20 


968766 


2279 


031234 


41 


2(1 


968249 


2252 


998116 


20 


970133 


2271 


029867 


40 


2) 


81)69600 


2244 


9-998104 


20 


8-971496 


2265 


11028504 


39 


22 


970947 


2238 


998092 


20 


972855 


2257 


027145 


38 


2:» 


972289 


2-231 


998080 


20 


974209 


2251 


025791 


37 


^ 


973628 


2224 


998068 


20 


975560 


2244 


024440 


36 


2.-. 


974962 


2217 


998056 


20 


976906 


2237 


023094 


35 


2<i 


976293 


2210 


998044 


20 


978248 


2230 


021752 


34 


27 


977619 


22(t3 


998032 


23 


979586 


2223 


0-20414 


33 


2H 


978941 


2197 


998020 


20 


980921 


2217 


019079 


32 


29 


980259 


2i90 


998008 


20 


982251 


2210 


017749 


31 


30 


981573 


2183 


997996 


20 


983577 


2204 


016423 


30 


31 


8-982883 


2177 


9-997984 


20 


8-984899 


2197 


11015101 


29 


32 


984189 


2170 


997972 


20 


98(5217 


2191 


013783 


28 


33 


985491 


2163 


997959 


20 


987532 


2184 


0124(^8 


27 


34 


986789 


2157 


997947 


20 


988842 


2178 


011158 


26 


35 


988083 


2150 


997935 


21 


990149 


2171 


009851 


25 


3(> 


989374 


2144 


997922 


21 


991451 


2165 


008549 


24 


37 


990660 


2138 


997910 


21 


992750 


2158 


007250 


23 


:<8 


991943 


2131 


997897 


21 


994045 


2152 


005955 


22 


39 


993222 


2125 


997885 


21 


995337 


2146 


00461)3 


2I 


4U 


994497 


2119 


997872 


21 


9966-24 


2140 


003376 


20 


41 


8-995768 


2112 


9-997860 


21 


8-997908 


2134 


11-002092 


19 


42 


997036 


2106 


997847 


21 


999188 


2127 


000812 


18 


43 


998299 


2100 


997835 


21 


9-000465 


2121 


10-999535 


17 


44 


999560 


2094 


997822 


21 


001738 


2115 


998262 


16 


45 


9-000816 


2087 


997809 


21 


003007 


2109 


996993 


15 


46 


002069 


2082 


997797 


21 


004272 


2103 


995728 


14 


47 


003318 


2076 


ttc)77«4 


21 


005.^34 


2097 


994466 


13 


48 


004563 


2070 


997771 


21 


006792 


2091 


993208 


12 


49 


005805 


2064 


997758 


21 


008047 


2085 


991953 


11 


50 


007041 


2058 


997745 


21 


009298 


2080 


990702 


10 


51 


9-008278 


2052 


9-997732 


21 


9 -01 0546 


2074 


10-989454 


9 


52 


009510 


2046 


997719 


21 


01J7i»0 


2068 


988210 


8 


53 


010737 


2040 


997706 


21 


013031 


2062 


986969 


7 


54 


011962 


2034 


997693 


22 


014-268 


2056 


985732 


6 


55 


013182 


2029 


997680 


22 


015502 


2051 


984498 


5 


56 


014400 


20-23 


997667 


22 


016732 


2045 


983268 


4 


57 


015613 


2017 


997654 


22 


017959 


2040 


982041 


3 


58 


016824 


•2012 


997641 


22 


019183 


2033 


980817 


2 


59 


018031 


2006 


997628 


22 


020403 


2028 


979597 


1 


60_ 


019235 


2000 


997614 


22 


021620 


2023 


978380 







Cnnine 




Sine 

84 


Degre 


1 Cotang. 

es. 


' 


1 Tang. 1 


M. 



204 (6 Degrees.) LOGARITILMIC iSINES, COSINES, ETC. 



H. 


1 Sine 


1 D. 


1 Cosine 


I D. 


1 Tang. 


1 D. 


Coia.i^. 







9-019235 


2000 


9997614 


22 


9-021620 • 


2023 


10-978380 


; 60 


1 


020435 


1995 


997601 


22 


0-2-2834 


2017 


977 KiO 


:>9 


2 


0'21ti:i2 


1989 


9i 17588 


22 


024044 


2011 


975956 


58 


3 


02-2H^25 


1984 


997574 


22 


0-25251 


2006 


974749 


57 


4 


<)'24(»I6 


1978 


997561 


22 


026455 


2.000 


973545 


56 


a 


0-25203 


1973 


997547 


22 


027 (i55 


1995 


972345 


55 


6 


()26:W6 


1967 


997534 


23 


028852 


1990 


971148 


54 


7 


027567 


1962 


997520 


23 


030046 


1985 


969954 


53 


J 


i<->i744 


1957 


997507 


23 


031-237 


1979 


9()8763 


52 


9 


(i-Jj<tll8 


1951 


997493 


23 


0:i2425 


1974 


967575 


51 


10 


(i;{|liH9 


1947 


997480 


23 


033609 


1969 


966391 


50 


11 


9(132257 


1941 


9-997466 


23 


9-034791 


1964 


10-965209 


49 


12 


ii:<;i42i 


193() 


997452 


23 


035969 


1958 


964031 


48 


13 


034582 


1930 


997439 


23 


037144 


1953 


962856 


47 


14 


035741 


1925 


997425 


23 


038316 


1948 


961(J84 


46 


15 


03H896 


1920 


997411 


23 


039485 


1943 


960515 


45 


16 


038048 


HI 15 


997397 


23 


040651 


1938 


959349 


44 


17 


039197 


1910 


997383 


23 


041813 


1933 


958187 


43 


18 


04u:<42 


iH(i5 


997369 


23 


042973 


1928 


957027 


42 


19 


041485 


1899 


997355 


23 


044130 


1923 


955870 


41 


2U 


042025 


1894 


997341 


23 


045284 


1918 


954716 


40 


21 


9 (M3762 


1889 


9-997327 


24 


9-046434 


1913 


10-953566 


39 


22 


044895 


1884 


997313 


24 


047582 


1908 


952418 


38 


23 


046026 


1879 


997299 


24 


048727 


1903 


951273 


37 


24 


047154 


1875 


997285 


24 


049869 


1898 


950131 


36 


25 


048279 


1870 


997271 


24 


051008 


1893 


948992 


35 


26 


049400 


1865 


997257 


24 


052144 


1889 


947856 


34 


27 


050519 


18C)0 


997242 


24 


053277 


1884 


946723 


33 


28 


051035 


1855 


9972-23 


24 


054407 


1879 


945593 


32 


29 


052749 


1850 


997214 


24 


055535 


187 i 


944465 


31 


30 


053859 


1845 


997199 


24 


056659 


1870 


943341 


30 


31 


054966 


1841 


9-997185 


24 


9-057781 


1865 


10-942219 


29 


32 


05()071 


1836 


997170 


24 


058900 


1869 


941100 


28 


33 


057172 


1831 


997156 


24 


060016 


1855 


939984 


27 


34 


058271 


1827 


997141 


24 


061130 


1851 


938870 


26 


35 


059367 


1822 


997127 


24 


062240 


1846 


937760 


25 


36 


060460 


1817 


997112 


24 


063348 


1842 


936652 


24 


37 


061551 


1813 


997098 


24 


064453 


1837 


935547 


23 


38 


062639 


1808 


997083 


25 


065556 


1833 


934444 


22 


39 


0()3724 


1804 


997068 


25 


066655 


1828 


833345 


21 


40 


064806 


1799 


997053 


25 


007752 


1824 


932248 


20 


41 


9'(,65R85 


1794 


9 997039 


25 


9-068846 


1819 


10-931154 


19 


42 


0«;6962 


1790 


9!)7024 


25 


069938 


1815 


930062 


18 


43 


068036 


17W6 


997009 


25 


071027 


1810 


928973 


17 


44 


069107 


1781 


996994 


25 


072113 


1806 


927887 


16 


45 


070176 


1777 


996979 


25 


073197 


1802 


926803 


15 


40 


071242 


1772 


996964 


25 


074278 


1797 


925722 


14 


47 


072:106 


1768 


99<i949 


25 


075356 


1793 


924644 


13 


48 


073366 


1763 


996934 


25 


(I7()432 


1789 


923568 


12 


49 


074424 


1759 


996919 


25 


077505 


1784 


922495 


11 


50 


075480 


1755 


996904 


25 


078576 


1780 


921424 


10 


51 


9076533 


1750 


9-99(1889 


25 


9-079644 


1776 


10-920356 


9 


5'J 


077583 


1746 


996874 


25 


080710 


1772 


919290 


8 


53 


078631 


1742 


99(5858 


25 


081773 


1767 


918227 


7 


54 


079676 


1738 


996843 


25 


082833 


1763 


917167 


6 


55 


080719 


1733 


996828 


25 


083891 


1759 


916109 


5 


56 


081759 


1729 


996812 


26 


084947 


1755 


915053 


4 


57 


082797 


1725 


99()797 


26 


08(i000 


1751 


914000 


3 


58 


0838112 


1721 


996782 


26 


087050 


1747 


912950 


2 


59 


0848(i4 


1717 


996766 


26 


088098 


1743 


911902 


1 


60 


085894 


1713 


996751 


26 


089144 


1738 


910856 







Coeine > 




Sme 


Degre 


Cotang. 

es. 


I 


Tang. I 


k 



LOGARITHMIC SINES, CO SIXES, ETC. (7 Degrees.; 



M 


1 Sine 


1 ^• 


1 Cosine 


1 D. 


1 Tan?. 


1 D. 


1 Cotan^ 


1 


~0 


9-085894 


1713 


9-996751 


26 


9-089144 


1738 


10-910856 


60 


1 


((H(iil-22 


1709 


<)96735 


26 


090187 


1734 


909813 


59 


2 


087947 


1704 


996720 


26 


091228 


1730 


908772 


58 


3 


088970 


1700 


99()704 


26 


092266 


1727 


907734 


.57 


4 


089990 


1696 


996688 


26 


093302 


1722 


'J()(S(!98 


56 


5 


091(108 


1()92 


99667:5 


26 


094336 


1719 


9()5()64 


55 


6 


092024 


](i88 


996657 


26 


095367 


1715 


904633 


.54 


7 


0930S7 


lii84 


99()(»41 


26 


096395 


1711 


903605 


53 


8 


094047 


Jfi80 


99()625 


96 


097422 


1707 


902578 


52 


9 


0950.')() 


l(i76 


996610 


26 


098446 


1703 


901554 


51 


10 


09(i0()2 


1(573 


996594 


26 


099468 


1699 


900532 


50 


11 


9-097(inr) 


KiCS 


9-99657S 


o- 


9-J0(!487 


1695 


10-899513 


49 


1-2 


O'jSOOli 


]i;(i-) 


!)!)(;.">( 12 


07 


101504 


1691 


898496 


48 


13 


0990(55 


]ii(ii 


i (ill ;.-,-( 6 


27 


102519 


1687 


897481 


47 


14 


1000()2 


Hi. -.7 


!t!l(i.");<0 


27 


103532 


1684 


896468 


46 


15 


10105(> 


i(;:)3 


H'.Ki.'iU 


"7 


104542 


I68ii 


895458 


45 


16 


102048 


l(i49 


it! MM! (8 


27 


105550 


1676 


894450 


44 


17 


103037 


l(i45 


9!l(;-182 


27 


J 06556 


1672 


893444 


43 


■18 


104025 


](i41 


9!Mi4(i5 


27 


I07o59 


i(;6<j 


89244 1 


42 


19 


105010 


1638 


9lt(i449 


27 


108560 


1665 


891440 


41 


20 


105992 


1634 


9;i(i433 


27 


109559 


1661 


890441 


40 


21 


9I(!()973 


]()30 


9-996417 


27 


9-110556 


1658 


10-889444 


39 


22 


107951 


1627 


99()4iiO 


07 


111551 


1654 


888449 


38 


23 


108027 


1623 


996384 


27 


1 12543 


1650 


887457 


37 


24 


i(ii»y:M 


16M! 


996368 




113533 


1646 


8864(57 


36 


25 


lios7:{ 


1616 


996351 


27 


114521 


1643 


885479 


35 


26 


J 1 1842 


1612 


996335 


27 


115507 


1639 


884493 


34 


27 


1 128(|i) 


1608 


996318 


07 


116491 


1636 


88351)9 


33 


2H 


113774 


1605 


996302 


28 


117472 


1632 


882528 


32 


29 


114737 


1601 


996285 


28 


1 18452 


1629 


881.548 


31 


31) 


115698 


1597 


99621)9 


28 


119429 


1625 


880571 


30 


31 


9 II6G56 


1594 


9-996252 


28 


9-120404 


1622 


10-879596 


29 


32 


117613 


1590 


996235 


28 


121377 


1()I8 


878623 


28 


33 


11S367 


1587 


996219 


28 


122348 


1615 


^^776.52 


27 


34 


119519 


1583 


996202 


28 


123317 


1611 


876683 


26 


35 


1204G9 


1580 


996185 


23 


124284 


I6(.'7 


875716 


25 


36 


121417 


1576 


996168 


28 


125249 


i()()4 


874751 


24 


37 


122302 


1573 


996151 


28 


126211 


KiOl 


S7:5789 


23 


38 


123306 


1569 


998134 


28 


127172 


1507 


872828 


22 


39 


124248 


1566 


996117 


28 


128130 


15111 


871870 


21 


40 


125 J 87 


1562 


996100 


28 


129087 


15!H 


870913 


20 


41 


912G125 


1559 


9-996083 


29 


9-130041 


1587 


10-8(59959 


19 


42 


1270(30 


1556 


996066 


29 


130994 


1584 


869006 


18 


43 


127993 


1552 


996049 


29 


131944 


1581 


868056 


17 


44 


128925 


1549 


996032 


29 


132893 


1577 


867107 


16 


45 


129854 


1545 


996015 


29 


133839 


1574 


8(56161 


15 


46 


130781 


1542 


995998 


29 


134784 


1571 


86.5216 


14 


47 


131706 


1539 


995980 


29 


135726 


15(57 


864274 


13 


48 


132630 


1535 


995963 


29 


136667 


15(54 


863333 


12 


49 


133551 


1532 


995946 


29 


137605 


1561 


862395 


11 


50 


134470 


1529 


995928 


29 


138542 


1558 


861458 


10 


51 


9135387 


1525 


9-995911 


29 


9139476 


1555 


10-860.524 


9 


52 


I3()303 


1522 


995894 


29 


140409 


1551 


8.59591 


8 


53 


137216 


1519 


H95876 


29 


141340 


1548 


858660 


7 


54 


138128 


1516 


995859 


29 


142269 


1545 


857731 


6 


55 


139037 


1512 


995841 


29 


143196 


1542 


856804 


5 


56 


139944 


1509 


995823 


29 


144121 


1539 


85.5879 


4 


57 


140850 


1506 


995806 


29 


145044 


1535 


854956 


3 


58 


141754 


1503 


995788 


29 


145966 


1532 


854034 


2 


59 


142655 


1500 


995771 


29 


146885 


1529 


853115 


1 


60 


143555 


1496 


995753 


29 


147803 


1526 


852197 






Sine I I Coumg'. | 



T««. IM. 



206 (8 Degrees.) LOGAIUTITMIC SINES, COSINES, ETC. 



M. 


1 Sine 


D. 


Cosine 


1 D. 


Tan<r. 


1 D. 


Cotang. 


1 





9-143555 


1496 


9 995753 


30 


9- 147803 


1526 


10.852197 


60 


1 


144453 


1493 


995735 


30 


148718 


1523 


851282 


59 


2 


145349 


1490 


995717 


3.) 


149632 


1520 


850368 


58 


3 


146243 


1487 


99569J 


30 


150544 


1517 


849456 


57 


4 


147136 


1484 


995681 


30 


151454 


1514 


848546 


56 


5 


148026 


1481 


995()(J4 


30 


152363 


1511 


847637 


55 


6 


148915 


1478 


993646 


30 


153269 


1598 


846731 


54. 


7 


149802 


1475 


995628 


30 


154174 


1505 


845826 


53 


8 


150G86 


1472 


995610 


30 


155077 


1502 


844<)-23 


52 


9 


151509 


1469 


995591 


30 


155978 


1499 


844022 


51 


10 


15'J451 


1466 


995573 


30 


• 156877 


1496 


843123 


50 


11 


9153330 


1463 


9-995555 


30 


9- 15/775 


1493 


10-842225 


49 


12 


154208 


1460 


995537 


30 


158671 


1490 


841329 


48 


13 


155083 


1457 


995519 


30 


159565 


1487 


840435 


47 


14 


155957 


1454 


995501 


31 


160457 


1484 


839543 


46 


15 


J 56830 


1451 


995482 


31 


161347 


1481 


838653 


45 


16 


157700 


1448 


995464 


31 


16-2-236 


1479 


837764 


44 


17 


158569 


1445 


995446 


31 


l(i3123 


1476 


836877 


43 


18 


159435 


1442 


995427 


31 


164008 


1473 


835992 


42 


19 


160301 


1439 


995409 


31 


1048!)2 


1470 


8351U8 


41 


20 


161164 


1436 


995390 


31 


165774 


1467 


834226 


40 


21 


9- J 62025 


1433 


9-995372 


31 


9' 166654 


1464 


10-833346 


39 


22 


J 62885 


1430 


995353 


31 


J 67532 


1461 


83-2468 


38 


23 


163743 


1427 


9it5334 


31 


168409 


1458 


8:!1.')91 


37 


24 


164600 


1424 


9953 M) 


31 


169284 


1455 


830716 


36 


25 


J 65454 


1422 


995297 


31 


170157 


1453 


82L»84.3 


35 


26 


J6()307 


1419 


995278 


31 


171029 


1450 


828971 


34 


27 


167159 


1416 


995260 


31 


171899 


1447 


828101 


33 


28 


J 68008 


1413 


995241 


32 


172767 


1444 


827233 


32 


29 


168856 


1410 


995222 


32 


173634 


1442 


82636() 


31 


30 


169702 


1407 


995203 


32 


174499 


1439 


825501 


30 


31 


9-170547 


1405 


9-995184 


32 


9-175362 


1436 


10-824638 


29 


32 


17l3Hi» 


1402 


995165 


32 


176224 


1433 


823776 


28 


33 


11^2-m 


139y 


995146 


32 


177(,y4 


1431 


822916 


27 


34 


173U70 


139(5 


995127 


32 


177942 


1428 


822058 


2(5 


35 


173908 


1394 


995108 


32 


178799 


J425 


821201 


25 


36 


174744 


1391 


995089 


32 


179655 


1423 


820345 


24 


37 


175578 


1388 


995070 


32 


180508 


1420 


819492 


23 


38 


176411 


1386 


995051 


32 


181360 


1417 


818640 


22 


39 


177242 


1383 


995032 


32 


182211 


1415 


817789 


21 


40 


178072 


1380 


995013 


32 


183059 


1412 


816941 


20 


41 


9-178900 


1377 


9-994993 


32 


9-183907 


1409 


10-81()093 


19 


42 


179726 


1374 


994974 


32 


184752 


1407 


81.^)248 


18 


43 


180551 


1372 


994955 


32 


185597 


1404 


814403 


17 


44 


181374 


1369 


994935 


32 


18()439 


1402 


81.3561 


16 


45 


182196 


1366 


994916 


33 


187280 


1309 


812720 


15 


46 


1830 J 6 


1364 


994896 


33 


188120 


1396 


811880 


14 


47 


183834 


1361 


994877 


33 


188958 


1393 


811042 


13 


48 


184651 


1359 


994857 


33 


189794 


1301 


810206 


12 


49 


185466 


1356 


994838 


33 


190629 


1389 


809371 


11 


50 


186280 


1353 


994818 


33 


101462 


1386 


808.538 


10 


51 


9-187092 


1351 


9-994798 


33 


S»' 192294 


1384 


l()-8077()6 


9 


52 


187903 


1348 


994779 


33 


193124 


1381 


806876 


8 


53 


188712 


1346 


994759 


33 


193953 


1379 


806047 


7 


54 


189519 


1343 


994739 


33 


194780 


1370 


805220 


6 


55 


190325 


1341 


994719 


33 


I956(K) 


1374 


804394 


5 


56 


191130 


1338 


994709 


33 


106430 


1371 


803570 


4 


57 


191933 


1336 


994()80 


33 


197253 


13()9 


802747 


2 


58 


192734 


1333 


994660 


33 


198074 


1366 


801926 


2 


59 


193534 


1330 


994640 


33 


198804 


1364 


801106 


1 


60 


194332 


1328 


994620 


33 


199713 


1361 


800287 






I Cosine | 



I Cotang-. I 



I Tang. 



81 Degrees. 



LOGARITHMIC S/XJ;S, COSfXES, ETC. (9 Degrofs.) 20^ 



M. 


Sine 


1 D- 


Cosine 


D. 


Taner. 
9-199713 


D. 
1361 


Colansr. 

TM)-S00-28y~ 




~0 


9194332 


1328 


9-994620 


33 


" 


1 


195129 


132() 


994600 


33 


200529 


1359 


799471 


i>9 


2 


195925 


1323 


994580 


33 


201345 


135() 


798655 


5" 


3 


196719 


1321 


994560 


34 


202159 


1354 


797841 


57 


4 


197511 


1318 


994540 


34 


202971 


1352 


797029 


5fi 


5 


198302 


1310 


994519 


34 


203782 


1349 


79(:218 


55 


6 


199091 


1313 


994499 


34 


204592 


1347 


795408 


54 


7 


199879 


1311 


994479 


34 


205400 


1345 


794(;00 


53 


8 


200(H)f) 


1308 


994459 


34 


206207 


1342 


793793 


52 


9 


201451 


1306 


994438 


34 


2070 IS 


1340 


792987 


51 


10 


202234 


1304 


994418 


34 


207817 


1338 


792183 


5«:' 


]l 


9-203017 


1301 


9-994397 


34 


9-208619 


1335 


10-791381 


49 


li 


203797 


1299 


994377 


34 


209420 


1333 


790580 


48 


i:i 


204577 


1296 


994357 


34 


210220 


1331 


789780 


47 


14 


205354 


1294 


99433() 


34 


211018 


1328 


788982 


46 


15 


20G131 


J 292 


994316 


34 


211815 


1326 


788185 


45 


l(i 


206906 


1289 


994295 


34 


212611 


1324 


787389 


44 


17 


207679 


1287 


994274 


35 


213405 


1321 


786595 


43 


18 


2*ia452 


1285 


994254 


35 


214198 


1319 


■; 85802 


42 


19 


209222 


1282 


994233 


35 


214989 


1317 


785011 


41 


20 


209992 


1280 


994212 


35 


215780 


1315 


784220 


40 


21 


9-210760 


1278 


9-994191 


35 


9-216568 


1312 


10-783432 


39 


22 


211526 


1275 


994171 


35 


217356 


1310 


782644 


38 


23 


212291 


1273 


994150 


35 


218142 


1308 


781858 


37 


24 


213055 


1271 


994129 


35 


218926 


1305 


781074 


36 


25 


213818 


1268 


994108 


35 


219710 


1303 


780290 


35 


2(5 


214579 


1266 


994087 


35 


220492 


1301 


779508 


34 


27 


215338 


1264 


994066 


35 


221272 


1299 


778728 


33 


28 


216097 


1261 


99^045 


35 


222052 


1297 


777948 


32 


29 


216854 


1259 


994024 


35 


222830 


1294 


777170 


31 


30 


217609 


1257 


994003 


35 


223606 


1292 


776394 


30 


31 


9-218363 


1255 


9-993981 


35 


9-224382 


1290 


10-775618 


29 


32 


219116 


1253 


993960 


35 


225156 


1288 


774844 


28 


33 


219868 


1250 


993939 


35 


225929 


1286 


774071 


27 


34 


220618 


1248 


993918 


35 


2-26700 


1284 


773300 


26 


35 


221367 


1246 


993896 


36 


227471 


1281 


772529 


25 


30 


222115 


1244 


993875 


36 


228239 


1279 


771761 


24 


37 


2-?.-?S61 


1242 


9D3854 


36 


229007 


1277 


770993 


23 


38 


223606 


1239 


993832 


36 


229773 


1275 


770227 


22 


39 


224349 


1237 


993811 


36 


230539 


1273 


769461 


21 


40 


225092 


1235 


993789 


36 


231302 


1271 


768698 


20 


41 


9-225833 


1233 


9-993768 


36 


9-232065 


1269 


10-707935 


19 


42 


22G573 


1231 


993746 


36 


232826 


lt:67 


767174 


18 


43 


227311 


1228 


993725 


;;(> 


233586 


l-i65 


766414 


17 


44 


228048 


1226 


993703 


36 


2::4;;45 


1262 


765655 


16 


45 


228784 


1224 


993681 


36 


2;-r>i;;3 


1260 


764897 


15 


46 


229518 


2020 


993660 


3ii 


2;:5t59 


1258 


764141 


11 


47 


230252 


1220 


993638 


36 


2::6(;i4 


1256 


763386 


13 


48 


230984 


1218 


993616 


36 


2:;73G8 


1254 


762632 


12 


49 


231714 


12i() 


993594 


!i7 


2:;8i2u 


1252 


761889 


11 


50 


232444 


1>J4 


993572 


37 


238872 


l-:50 


761128 


10 


51 


9-233172 


1212 


9-993550 


37 


'.)-2n:ir,22 


1248 


10-760378 


9 


52 


233899 


■ 1209 


9935-28 


37 


24u::7i 


124(i 


759629 


8 


53 


234625 


1207 


993506 


37 


241118 


m4- 


758882 


7 


54 


235349 


1205 


993484 


37 


241865 


1242 


758135 


e 


55 


236073 


1203 


993462 


37 


2426W 


1240 


757390 


5 


5{) 


236795 


1201 


993440 


37 


243354 


1238 


756646 


4 


57 


237515 


1199 


993418 


37 


244097 


1236 


755903 


3 


5>) 


23e2;!5 


1197 


993396 


37 


244839 


1234 


755161 


2 


59 


238953 


1195 


993374 


37 


^5579 


1232 


754421 1 


i 


GU 


239670 


1193 


993351 


37 


246319 


1230 


753681 







Cosine ! 


1 


Sme 1 
80 


Degre 


Cotaii^. 
3S. 




Tang. 1 


M. 



208 (10 D<'grees.) LOGARITHMIC SINES, COSINES, ETC. 



M. 


1 Sine 


i D. 


1 Cosine 


I D. 


f Tan^. 


1 D. 


1 Cotang. 




o' 


9-239670 


1193 


9-993351 


37 


9-246319 


1230 


10-75G681 


60 


1 


240386 


1191 


993329 


37 


247057 


1228 


752943 


59 


2 


241101 


1189 


993307 


37 


247794 


1-226 


752206 


58 


3 


241814 


1187 


993285 


37 


248530 


1224 


751470 


57 


4 


242526 


1185 


993262 


37 


249264 


1222 


750736 


56 


5 


243237 


1183 


993240 


37 


249998 


]2-2d 


750002 


55 


6 


243947 


J 181 


993217 


38 


250730 


1218 


749270 


54 


7 


244656 


1179 


993195 


38 


251461 


1217 


748539 


53 


8 


245363 


1177 


993172 


38 


252191 


1215 


747809 


52 


9 


246069 


1175 


993149 


38 


252920 


1213 


747080 


51 


10 


246775. 


1173 


993127 


38 


253648 


1211 


746352 


50 


11 


9-247478 


1171 


9-993104 


38 


9-254374 


1209 


10-745G26 


49 


12 


248181 


IIGU 


993081 


38 


255100 


1207 


744J00 


48 


13 


248883 


1107 


993059 


38 


255824 


1205 


744176 


47 


14 


249583 


1165 


993036 


38 


256547 


1203 


743453 


46 


15 


230282 


1163 


993013 


38 


257269 


1201 


742731 


45 


16 


250980 


1161 


992990 


38 


257990 


1200 


742010 


44 


17 


251677 


1159 


992967 


38 


258710 


1198 


741290 


43 


18 


252373 


1158 


992944 


38 


259429 


1196 


740571 


42 


19 


253067 


1156 


992921 


38 


260146 


1194 


739854 


41 


20 


253761 


1154 


992898 


38 


260863 


1192 


739137 


40 


21 


9-254453 


1152 


9-992875 


38 


9-261578 


1190 


10-738422 


39 


22 


255144 


1150 


99-2852 


38 


262292 


1189 


737708 


38 


23 


255834 


1148 


992829 


39 


263005 


1187 


736995 


37 


24 


256523 


1146 


992806 


39 


2G3717 


1185 


73G283 


36 


25 


257211 


1144 


992783 


39 


2G4428 


1183 


735572 


35 


26 


257898 


1142 


992759 


39 


2G5138 


1181 


734862 


34 


27 


258583 


1141 


992736 


39 


265847 


1179 


734153 


33 


28 


259268 


1139 


992713 


39 


26o555 


1178 


733445 


32 


29 


259951 


1137 


992690 


39 


2672G1 


117G 


732739 


31 


30 


260633 


1135 


992606 


39 


2G7967 


1174 


732033 


30 


31 


9-261314 


1133 


9-992643 


39 


9-268671 


1172 


10-7313-29 


29 


32 


261994 


1131 


992619 


39 


2G9375 


1170 


730625 


28 


33 


262673 


1130 


992596 


39 


270077 


11G9 


729923 


27 


34 


203351 


1128 


992572 


39 


270779 


1167 


729221 


26 


35 


2G4027 


1126 


992549 


39 


271479 


11G5 


728521 


25 


36 


264703 


1124 


992525 


39 


272178 


1164 


727822 


24 


37 


265377 


1122 


992501 


39 


272876 


1162 


727124 


23 


38 


2G6051 


1120 


992478 


40 


273573 


IIGO 


726427 


S 


39 


266723 


1119 


992454 


40 


274269 


1158 


725731 


21 


40 


267395 


1117 


992430 


40 


274964 


1157 


725036 


20 


41 


9-268065 


1J15 


9-992406 


40 


9-275G58 


1155 


10-724342 


19 


42 


268734 


1113 


992382 


40 


276351 


1153 


723649 


18 


43 


2G9402 


1111 


992359 


40 


277043 


1151 


7t>2957 


17 


44 


270069 


1110 


992335 


40 


277734 


1150 


722266 


16 


45 


270735 


1108 


992311 


40 


278424 


1148 


721576 


15 


46 


271400 


1106 


992287 


40 


279113 


1147 


720887 


14 


47 


272064 


1105 


9922G3 


40 


279801 


1145 


720199 


13 


48 


272726 


1103 


992239 


40 


280488 


1143 


719512 


12 


49 


273388 


1101 


992214 


40 


281174 


1141 


718826 


11 


50 


274049 


1099 


992190 


40 


281858 


1140 


718142 


10 


51 


9-274708 


1098 


9-992166 


40 


9-282542 


1138 


10-717458 


9 


52 


275367 


1096 


992142 


40 


283225 


1136 


716775 


8 


53 


276024 


1094 


992117 




283907 


1135 


716093 


7 


54 


276681 


1092 


992093 




284588 


1133 


715412 


6 


55 


277337 


1091 


992069 




2852G8 


1131 


714732 


5 


56 


277991 


1089 


992044 




285947 


1130 


714053 


4 


57 


278644 


1087 


992020 




286G24 


1128 


713376 




58 


279297 


1086 


991996 




287301 


1126 


712699 


"i 


59 


279948 


1084 


991971 




287977 


1125 


712023 


1 


60 


280599 


1082 


991947 




288652 


1123 


711348 


-9 


1 


Cosine 


1 


Sine 

79 


1 
Degree 


Cotang. 1 


1 


Tang. 1 


K. 



LOGARITHMIC 8INE8, COSINES, ETC. C^l Degrees.) 201' 



.M. 


1 Sins 


D. 


Cosine 


D. 


Tang. 


D. 


I Cotang. 







9-280599 


1082 


9-991947 


41 


9-288652 


1123 


10-711348 


60 


1 


281248 


1081 


991922 


41 


289326 


1122 


710674 


59 


2 


281897 


1079 


991897 


41 


289999 


1120 


710001 


58 


3 


282544 


1077 


991873 


41 


290671 


1118 


709329 


57 


4 


283190 


1076 


991848 


41 


291342 


1117 


708658 


56 


5 


283836 


1074 


991823 


41 


292013 


1115 


707987 


55 


fi 


2«4480 


1072 


991799 


41 


292682 


1114 


707318 


54 


- 


2-S5I24 


107 1 


991774 


42 


293350 


1112 


706650 


53 


8 


285766 


1069 


991749 


42 


294017 


1111 


705983 


52 


9 


286408 


1067 


991724 


42 


294684 


1109 


70.5316 


51 


10 


287048 


1066 


991099 


42 


295349 


1107 


^04651 


50 


11 


9-287687 


1064 


9-991674 


42 


9-296013 


1106 


10-703987 


49 


12 


288326 


1063 


991649 


42 


296677 


1104 


703323 


48 


13 


288964 


1061 


991624 


- 42 


297339 


1103 


702661 


47 


14 


289600 


1059 


991599 


42 


298001 


1101 


701999 


46 


15 


290236 


1058 


991574 


42 


298662 


1100 


7013.38 


45 


10 


290870 


1056 


991549 


42 


299322 


1098 


700678 


44 


17 


291504 


1054 


991524 


42 


299980 


1096 


700020 


43 


le 


292137 


1053 


991498 


42 


300638 


1095 


699362 


42 


19 


292768 


1051 


991473 


42 


301295 


1093 


698705 


41 


20 


293399 


1050 


991448 


42 


30I95I 


1092 


698049 


40 


21 


9-294029 


1048 


9-991422 


42 


9-302607 


1090 


10-697393 


39 


22 


294658 


1046 


991397 


42 


303261 


1089 


696739 


38 


23 


295286 


1045 


991372 


43 


303914 


1087 


696086 


37 


ai 


295913 


1043 


991346 


43 


304567 


1086 


695433 


36 


25 


296539 


1042 


991321 


43 


303218 


1084 


694782 


35 


26 


297164 


1040 


9D12J5 


43 


305869 


1083 


694131 


34 


27 


297788 


1039 


991270 


43 


306519 


1081 


693481 


33 


28 


298412 


1037 


991244 


43 


3071 08 


1080 


69'2832 


32 


29 


299034 


1036 


991218 


43 


307815 


1078 


692185 


31 


30 


2i)9655 


1034 


991193 


43 


308463 


1077 


691537 


30 


31 


9-300276 


1032 


9-991167 


43 


C-30Dlfir) 


1075 


10-690891 


29 


32 


300895 


1031 


991141 


43 


309754 


1074 


690246 


28 


33 


301514 


1029 


991115 


43 


310398 


1973 


689602 


27 


34 


302132 


1028 


991090 


43 


311042 


1071 


688958 


26 


35 


302748 


1028 


9910G4 


43 


311685 


1070 


688315 


25 


36 


303364 


1025 


991038 


43 


312327 


1068 


687673 


24 


37 


303979 


1023 


991012 


43 


312967 


1067 


687033 


23 


38 


304593 


1022 


990986 


43 


313608 


1065 


686392 


22 


39 


305207 


1020 


990960 


43 


314247 


1064 


685753 


21 


40 


305819 


1019 


990934 


44 


314885 


1062 


685115 


20 


41 


9-306430 


1017 


9-990908 


44 


9 315523 


1061 


10-684477 


19 


42 


307041 


1016 


990882 


44 


3161.59 


1060 


683841 


18 


43 


307650 


1014 


990855 


44 


316795 


1058 


683205 


17 


44 


308259 


1013 


990829 


44 


317430 


1057 


682570 


16 


45 


308867 


1011 


900803 


44 


318064 


1055 


681936 


15 


46 


n;)9474 


1010 


990777 


44 


218697 


1054 


681303 


14 


47 


310080 


1008 


9907.:0 


44 


319.329 


10.33 


680671 


13 


48 


310685 


1007 


990724 


44 


319961 


1051 


680039 


12 


49 


311289 


1005 


990697 


44 


320592 


1050 


679408 


11 


50 


311893 


1004 


99(1671 


44 


321222 


1048 


678778 


10 


5J 


9-212495 


1003 


9-990644 


44 


9-321851 


1047 


10-678149 


9 


52 


31309/ 


1001 


9906J8 


44 


322479 


1045 


G77.521 


8 


53 


313698 


1000 


990.591 


44 


3-231(i6 


1044 


676894 




54 


314297 


998 


990565 


44 


323733 


1043 


67G267 


6 


55 


214897 


997 


99J538 


44 


■.vl-2-.S 


1041 


675642 


5 


5«i 


31.5495 


996 


990511 


45 


^"24983 


1040 


675017 


4 


57 


SI 6092 


994 


990485 


45 


3-25C^7 


1039 


674393 


3 


58 


3Jf)6«9 


993 


990458 


45 


320231 


1037 


6727G9 


2 


59 


317284 


991 


990431 


45 


326853 


1036 


673147 


1 


60 


317879 


990 


990404 


45 


327475 


1035 


672525 







Cosine I 


I 


Sine 1 
78 


Degre* 


Cotang. 
)8. ' 




Tang. 1 


M. 



210 


(12 Ttri, 


rccs.) 


LOGARITHMIC 


SINES, 


COSLYES. ETC. 




M. 


Sine 


D. 


Cosine 


D. 


1 Tang. 


1 D. 


ColUMg. 




u" 


9-317879 


990 


9-990404 


45 


9-327474 


1035 


10-672^26 


60 


1 


318473 


988 


990378 


45 


328095 


1033 


671L05 


59 


2 


319066 


987 


990351 


45 


328715 


1032 


671285 


58 


3 


319658 


986 


990324 


45 


329334 


1030 


67066G 


57 


4 


320249 


984 


990297 


45 


329953 


1029 


670047 


56 


5 


320840 


983 


990270 


45 


330570 


1028 


669430 


55 


6 


321430 


982 


990243 


45 


331187 


1026 


668813 


54 I 


7 


322019 


980 


990215 


45 


331803 


1025 


668197 


53 


8 


322607 


979 


990188 


45 


332418 


1024 


667582 


52 


9 


323194 


977 


990161 


45 


333033 


1023 


666967 


51 


10 


323780 


976 


990134 


45 


333646 


1021 


666354 


50 


11 


9-3243()6 


975 


9-990107 


40 


9-334259 


1020 


10-605741 


49 


12 


32495* ) 


973 


990079 


40 


334871 


1019 


665129 


48 


13 


325534 


972 


990052 


46 


335482 


1017 


604518 


47 


J4 


326117 


970 


990U25 


40 


336093 


1016 


6639b'7 


40 


J5 


326700 


969 


989997 


46 


336702 


1015 


663298 


45 


1() 


32728J 


968 


989970 


46 


33731] 


1013 


662689 


44 


17 


3278(i2 


96G 


989942 


40 


337919 


1012 


6C2C81 


43 


18 


328442 


965 


989915 


46 


338527 


1011 


661473 


42 


19 


329021 


964 


989887 


46 


339133 


1010 


660867 


41 


20 


329599 


962 


9S98G0 


46 


339739 


1008 


660261 


40 


21 


9-330176 


961 


9-989832 


46 


9-340344 


1007 


[0-659656 


39 


22 


330753 


960 


989804 


40 


340948 


1006 


659052 


38 


23 


331329 


958 


989777 


46 


341552 


1004 


658448 


37 


24 


331903 


957 


989749 


47 


342155 


1003 


657845 


36 


25 


332478 


956 


989721 


47 


342757 


1002 


657243 


35 


20 


333051 


954 


989693 


47 


343358 


1000 


656642 


34 


27 


333624 


953 


989G65 


47 


343958 


999 


656042 


33 


28 


334195 


952 


989037 


47 


344558 


998 


655442 


32 


29 


334706 


950 


989609 


47 


345157 


997 


654843 


31 


30 


335337 


949 


989582 


47 


345755 


996 


654245 


30 


31 


9-335906 


948 


9-989553 


47 


9-346353 


994 


10-653647 


29 


32 


336475 


946 


989525 


47 


346949 


9:)3 


653051 


28 


33 


337043 


945 


989497 


47 


347545 


992 


652455 


27 


34 


337610 


944 


9894Ci9 


47 


348141 


991 


651859 


26 


35 


338176 


943 


989441 


47 


348735 


990 


651265 


25 


36 


338742 


941 


989413 


47 


349329 


988 


650671 


24 


37 


339306 


940 


989384 


47 


349922 


987 


650078 


23 j 


38 


339871 


939 


989356 


47 


350514 


986 


649486 


22 


39 


340434 


937 


98'.>328 


47 


35J106 


985 


648894 


21 


40 


340996 


936 


989300 


47 


351697 


983 


648303 


20 


41 


9-341558 


935 


9-989271 


47 


9-352287 


982 


10 647713 


19 


42 


342119 


934 


989243 


47 


352876 


981 


647124 


18 


43 


342(579 


932 


989214 


47 


353465 


980 


64(5535 


17 


44 


343239 


931 


989186 


47 


354053 


979 


645947 


16 


45 


343797 


930 


989157 


47 


354640 


977 


645360 


15 


46 


344355 


929 


989128 


48 


355227 


976 


644773 


14 


47 


344912 


927 


989100 


48 


355813 


975 


644187 


13 


48 


345469 


926 


989071 


43 


356398 


974 


643602 


12 


49 


346024 


925 


989042 


48 


356982 


973 


043018 


U 


50 


346579 


924 


989014 


48 


357566 


971 


642434 


10 


51 


9-347134 


922 


9-98ffii85 
988956 


48 


9-358149 


970 


10-641851 


9 


52 


347687 


921 


48 


358731 


969 


641260 


8 


53 


34824(1 


920 


988927 


48 


359313 


968 


(540(587 


7 


54 


348792 


919 


988898 


48 


359893 


967 


(i4()I07 


6 


55 


349343 


917 


988869 


48 


360474 


966 


639520 


5 


36 


349893 


916 


988840 


48 


361053 


965 


(538947 


4 


57 


350443 


915 


988811 


49 


361632 


9(i3 


63P3C.8 


3 


58 


350992 


914 


988782 


49 


362210 


962 


637790 


2 


59 


351540 


913 


988753 


49 


362787 


961 


637213 


1 


60 


352088 


911 


988724 


49 


363364 


960 


636636 







Cosine 




1 Sine 

T 


7 Degrt 


Cotang. 
»es. 




Tang. ; 


JVL 



LOQABITHMIG SINES, COSINES, ETC. (13 Degrees.) 211 



M. 


1 Sme 


1 D. 


Cosine 


1 D. 


Tan J. 


D. 


Cotang. 







9-352088 


911 


9-988724 


49 


9-363364 


960 


-^0-636636 


60 


1 


352635 


910 


988695 


49 


363940 


959 


636060 


59 


2 


353181 


909 


988666 


49 


364515 


958 


635485 


58 


3 


353726 


908 


988636 


49 


365090 


957 


634910 


57 


4 


354271 


907 


988607 


49 


365664 


955 


634336 


56 


5 


354815 


905 


988578 


49 


360237 


954 


633763 


55 


6 


355358 


904 


988548 


49 


366810 


953 


633190 


54 


7 


355901 


903 


988519 


49 


367382 


952 


632618 


53 


8 


356443 


902 


988489 


49 


367953 


951 


632047 


52 


9 


356984 


901 


988460 


49 


368524 


950 


631476 


51 


10 


357524 


899 


988430 


49 


309094 


949 


630906 


50 


11 


9-358064 


898 


9-988401 


49 


9-369663 


948 


10-630337 


49 


12 


358603 


897 


988371 


49 


370232 


946 


629768 


48 


13 


359J41 


896 


988342 


49 


370799 


945 


629201 


47 


14 


35U()78 


895 


988312 


50 


371367 


944 


628633 


46 


15 


360215 


893 


988282 


50 


371933 


943 


028067 


45 


16 


360752 


892 


988252 


50 


372499 


942 


027501 


44 


17 


361287 


891 


988223 


50 


373064 


941 


620936 


43 


18 


361822 


890 


988193 


50 


373629 


940 


626371 


42 


19 


362356 


889 


988163 


50 


374193 


939 


625807 


41 


20 


3!)2889 


888 


988133 


50 


374756 


938 


625244 


40 


21 


9-363422 


887 


9-988103 


50 


9-375319 


937 


10-624681 


39 


22 


3()3954 


885 


988073 


50 


375881 


935 


624119 


38 


23 


304485 


884 


988043 


50 


376442 


934 


623558 


37 


24 


365016 


883 


988013 


50 


377003 


933 


622997 


36 


25 


3(;5546 


882 


987983 


50 


377563 


932 


62-2437 


35 


26 


366075 


881 


987953 


50 


378122 


931 


621878 


34 


27 


366604 


880 


987922 


50 


378081 


930 


621319 


33 


28 


367131 


879 


987892 


50 


379239 


929 


020761 


32 


29 


3(;7n59 


877 


987862 


50 


379797 


928 


620203 


31 


30 


368185 


876 


987832 


51 


380354 


927 


619646 


30 


31 


9-368711 


875 


9-987801 


51 


9-380910 


926 


10-619090 


29 


32 


369236 


874 


987771 


51 


381466 


925 


618534 


28 


33 


369761 


873 


987740 


51 


382020 


924 


617980 


27 


34 


370285 


872 


987710 


51 


382575 


923 


617425 


26 


35 


370808 


871 


987679 


51 


383129 


922 


010871 


25 


36 


371330 


870 


987649 


51 


383682 


921 


616318 


24 


37 


371852 


869 


987618 


51 


384234 


920 


615766 


23 


38 


372373 


867 


987588 


51 


384786 


9J9 


615214 


22 


39 


372894 


866 


987557 


51 


385337 


918 


614663 


21 


40 


373414 


865 


987526 


51 


385888 


917 


014112 


20 


41 


9-373933 


804 


9-987496 


51 


9-3864:^8 


915 


10-013502 


19 


42 


374452 


863 


987465 


51 


380987 


914 


013013 


18 


43 


374970 


862 


987434 


51 


387536 


913 


012404 


17 


44 


375487 


861 


987403 


52 


388084 


912 


611916 


16 


45 


376003 


860 


987372 


52 


388631 


911 


61 1309 


15 


46 


376519 


859 


987341 


52 


389178 


910 


010822 


14 


47 


377035 


858 


987310 


52 


389724 


909 


610-276 


13 


48 


377549 


857 


987279 


52 


390270 


908 


609730 


12 


49 


378063 


856 


987248 


52 


•3908 15 


907 


609185 


11 


50 


378577 


854 


987217 


52 


391360 


906 


608640 


10 


51 


9-379089 


853 


9-987186 


52 


9-391903 


905 


10-608097 


9 


52 


379601 


852 


987155 


52 


392447 


904 


607553 


8 


53 


380113 


851 


987124 


52 


392989 


903 


607011 


7 


54 


380624 


850 


987092 


52 


393531 


902 


606409 


6 


55 


381134 


849 


987061 


52 


394073 


901 


005927 


5 


56 


381643 


848 


987030 


52 


394614 


900 


605386 


4 


57 


382152 


847 


986y98 


52 


395154 


899 


604846 


3 


58 


382661 


846 


986967 


52 


395094 


898 


604306 


2 


59 


383108 


845 


986936 


52 


396233 


897 


603767 


I 


60 


383675 


844 


• 986904 


52 


396771 


896 


603229 






I I Cotaug. 

76 Degrees. 



Taug. I 



212 (14 Degrees.) LOGARITHMIC SINES, COSINES, BTC 



I Tang. I D. I Cotang. | 






9-383675 


844 


9-986904 


52 


9-396771 


896 


10603229 


60 


I 


384182 


843 


986873 


53 


397309 


896 


602691 


59 


2 


384087 


842 


986841 


53 


397346 


895 


002154 


58 


3 


385 192 


841 


986809 


53 


398383 


894 


601617 


57 


4 


385G97 


840 


986778 


53 


398919 


SD3 


601(,81 


56 


5 


380231 


■^::9 


980740 


53 


;;;;j-^55 


892 


600545 


55 


G 


38G704 


b;>8 


986714 


53 


399990 


891 


600010 


54 


7 


387207 


837 


986683 


53 


400524 


890 


599476 


53 


.8 


387709 


836 


986651 


53 


401058 


889 


598942 


52 


9 


388210 


835 


980619 


53 


401591 


888 


598409 


51 


10 


388711 


834 


986587 


53 


402124 


887 


597876 


50 


11 


9-38921 1 


833 


9-986555 


53 


9-402050 


886 


10-597344 


49 


12 


389711 


832 


986523 


53 


403187 


885 


596813 


48 


13 


390210 


831 


986491 


53 


403718 


884 


596282 


47 


14 


390708 


830 


980459 


53 


404249 


883 


595751 


46 


15 


391206 


828 


980427 


53 


404778 


882 


595222 


45 


16 


391703 


827 


980395 


53 


405308 


881 


594692 


44 


17 


392199 


826 


986303 


54 


405836 


880 


594164 


43 


18 


392095 


825 


980331 


54 


406364 


879 


593636 


42 


19 


393191 


824 


980299 


54 


406892 


878 


503108 


41 


20 


393085 


823 


986206 


54 


407419 


877 


592581 


40 


21 


9-394179 


822 


9-986234 


54 


9-407945 


876 


10-592055 


39 


22 


394 (i73 


821 


986202 


54 


408471 


875 


591529 


38 


23 


395106 


820 


986109 


54 


408997 


874 


591003 


37 


24 


395058 


819 


980137 


54 


409521 


874 


590479 


36 


25 


390150 


818 


986104 


54 


410045 


873 


589955 


35 


26 


390041 


817 


980072 


54 


410509 


872 


589431 


34 


27 


397132 


817 


986039 


54 


411092 


871 


588908 


33 


28 


397021 


816 


986007 


54 


4U015 


870 


588385 


32 


29 


398111 


815 


985974 


54 


412137 


869 


587863 


31 


30 


398000 


814 


985942 


54 


412658 


868 


587342 


30 


31 


C-3990a8 


813 


9-985909 


55 


9-413179 


867 


10-586821 


29 


32 


399575 


812 


985876 


55 


413699 


806 


586301 


28 


33 


400062 


811 


985843 


55 


414219 


805 


585781 


27 


34 


400549 


810 


985811 


55 


414738 


864 


585262 


26 


35 


401035 


809 


985778 


55 


415257 


804 


584743 


25 


30 


401520 


808 


985745 


55 


415775 


803 


584225 


24 


37 


402005 


807 


985712 


55 


416293 


802 


583707 


23 


38 


402489 


806 


985679 


55 


416810 


8()1 


583190 


22 


39 


402972 


805 


985646 


55 


417320 


860 


582674 


21 


40 


403455 


804 


985613 


55 


417842 


859 


582158 


20 


41 


{)-403938 


803 


9-985580 


55 


9-418358 


858 


10-581642 


19 


42 


404420 


802 


985547 


55 


418873 


857 


581127 


18 


43 


404901 


801 


985514 


55 


419387 


856 


580613 


17 


44 


405382 


800 


985480 


55 


419901 


855 


580099 


16 


45 


405862 


799 


985447 


55 


420415 


855 


579585 


15 


46 


406341 


798 


985414 


56 


420927 


854 


579073 


14 


47 


406820 


797 


985380 


56 


421440 


853 


578560 


13 


48 


407299 


796 


985347 


56 


421952 


852 


578048 


12 


49 


407777 


795 


985314 


56 


422-463 


851 


577537 


11 


50 


408254- 


794 


985280 


56 


422974 


850 


577026 


10 


51 


9-408731 


794 


9-985247 


56 


9-423484 


849 


10-576516 


9 


52 


409207 


793 


985213 


50 


423993 


848 


576007 


8 


53 


409(«2 


792 


985180 


50 


424503 


848 


575497 


7 


54 


410157 


791 


985146 


56 


425011 


847 


574989 


6 


55 


410632 


790 


985113 


56 


425519 


846 


574481 


5 


56 


411106 


789 


985079 


56 


426027 


845 


573973 


4 


57 


411579 


788 


985045 


56 


426534 


844 


573466 


3 


58 


412052 


787 


985011 


56 


427041 


843 


572959 


2 


59 


412524 


786 


984978 


56 


427547 1 


843 


572453 


1 


60 


412996 


785 


984944 


56 


428052 1 


842 


571948 







Coiin» 


1 


Sine i 
75 


1 
Degre 


Cotang. I 

es. 




Taag. 1 


VL 



LOGARITHMIC SIXES, CO SIXES, ETC. (To Ix-grees.) 213 



M. ( 



D. I TaMR. 






9-412996 


1 785 


9-9R4944 


57 


9-428052 


842 


10-571948 


' 


1 


4i:!4fi7 


784 


9^4910 


57 


428557 


841 


571443 




2 


4i3y;;8 


1 783 


984876 


57 


429062 


840 


570938 




3 


414408 


783 


984842 


57 


429566 


839 


570434 




4 


414878 


782 


98-1808 


57 


430070 


838 


509930 




5 


415347 


781 


984774 


57 


430573 


838 


569427 




6 


415815 


780 


984740 


57 


431075 


837 


5C8925 




7 


416283 


779 


984706 


57 


431577 


836 


568423 




8 


410751 


778 


984672 


! 57 


432079 


835 


507921 




9 


4J7217 


777 


984637 


57 


432580 


834 


567420 




10 


■ 417G84 


776 


984603 


57 


433080 


833 


566920 




11 


9-418150 


775 


9-984569 


57 


9-433580 


832 


iO-5G0420 




12 


4J86I5 


774 


984535 


57 


434080 


832 


505920 




13 


419079 


773 


984500 


57 


434579 


831 


505421 


^ 


14 


410544 


773 


984466 


57 


435078 


830 


564922 




15 


420007 


772 


984432 


58 


435576 


829 


564424 




16 


420470 


771 


98-1397 


58 


436073 


828 


563927 




17 


420933 


770 


984363 


58 


436570 


828 


563430 




18 


421395 


769 


984328 


58 


437007 


827 


562933 


A 


^9 


42J857 


708 


984294 


58 


437563 


826 


562437 


I 


20 


4223J8 


707 


984259 


58 


438059 


825 


561941 




21 


9-422778 


767 


9-984224 


58 


9-438554 


824 


10-561446 


\_ 


22 


423238 


766 


984190 


58 


439048 


823 


560952 


[ 


23 


423697 


7C5 


984155 


58 


439543 


823 


560457 


[ 


24 


424156 


764 


984120 


58 


440036 


822 


559964 


r 


35 


424G15 


763 


984085 


58 


440529 


821 


559471 


; 


26 


425073 


762 


984050 


58 


441022 


820 


558978 


' 


27 


425530 


761 


984015 


58 


441514 


819 


558486 


2 


28 


425987 


700 


983981 


58 


442006 


819 


557994 


; 


29 


420443 


760 


983946 


58 


442497 


818 


557503 


3 


30 


426899 


759 


98391 1 


58 


442988 


817 


557012 


3 


31 


9-427354 


758 


9-983875 


58 


9-443479 


816 


10-556521 


2 


32 


427809 


757 


983840 


59 


443908 


816 


556032 





33 


428263 


756 


983805 


59 


444458 


815 


555542 


c 


34 


428717 


755 


983770 


59 


444947 


814 


555053 


2 


35 


429170 


754 


983735 


59 


445435 


813 


554565 


t_ 


36 


429023 


753 


983700 


59 


445923 


812 


554077 


2 


37 


430075 


752 


983664 


59 


446411 


812 


553580 





38 


430527 


752 


983629 


59 


446898 


811 


553103 


; 


39 


430978 


751 


983594 


59 


447334 


810 


552010 


.; 


40 


431429 


750 


983558 


59 


447870 


809 


552130 


2 


41 


9-431879 


749 


9-983523 


59 


9-448356 


809 


10-551644 


1 


42 


432329 


749 


983487 


59 


448841 


808 


551159 


1 


43 


432778 


748 


983452 


59 


44932G 


807 


550674 


I 


44 


433226 


747 


983416 


59 


449810 


806 


550190 


1 


45 


433G75 


746 


983381 


59 


450294 


806 


549706 


1 


4G 


434 J 22 


745 


983345 


59 


450777 


805 


549223 


1 


47 


434.169 


744 


983309 


59 


•451260 


804 


548740 


1 


4f 


435016 


744 


983273 


60 


451743 


803 


548257 


1 


4t 


435462 


743 


983238 


60 


452225 


802 


547775 


1 


5(, 


435908 


742 


983202 


60 


452706 


802 


547294 


1 


51 


9-436353 


741 


9-983l(!6 


60 


9-453187 


801 


10-546813 




52 


436798 


740 


983130 


60 


453668 


800 


546332 




53 


437242 


740 


983094 


60 


454148 


799 


545852 




54 


4376H6 


739 


983058 


60 


454628 


799 


545372 




55 


438129 


738 


983022 


60 


455107 


798 


544893 




56 


438572 


737 


982986 


60 


455586 


797 


544414 




57 


439014 


736 


982950 


60 


456064 


796 


543936 




58 


439456 


736 


982914 


60 


456542 


796 


543458 




59 


439897 


735 


982878 


60 


457019 


795 


542981 




60 


440338 


734 


982842 


60 


457496 


794 


542504 1 



Sine I 



I Cotang. I 



\ Tan§ 



74 Degrees. 



214 (10 DogifcK.; LOUARirilMIC SINES, COSIXES, ETC. 



M. 


Sine 
9-440338 


D. 
734 


Cosine 


1 D. 


1 Tan-. 


1 f>- 


1 Cotansr. 


1 


U 


~9~982842~ 


60 


9-457496 


7'J4 


10-542504 


60 


I 


440778 


733 


982805 


()0 


457973 


7^)3 


542027 


59 


2 


441218 


732 


982709 


61 


458449 


793 


541551 


58 


3 


441658 


731 


932733 


61 


458925 


792 


541075 


57 


4 


442096 


731 


982696 


61 


459400 


791 


540600 


56 


5 


442535 


730 


982660 


61 


459875 


790 


540125 


55 


G 


442973 


729 


982624 


61 


460349 


790 


539651 


51 


7 


443410 


728 


982587 


61 


460823 


789 


539177 


53 


8 


443847 


727 


982551 


61 


461297 


788 


538703 


52 


9 


444284 


727 


982514 


61 


461770 


788 


538230 


51 


10 


444720 


726 


982477 


01 


462242 


787 


537758 


50 


]1 


9-445155 


725 


9-982441 


61 


9-462714 


786 


10-537286 


49 


12 


445590 


724 


982404 


61 


463186 


785 


536814 


48 


13 


446025 


723 


9P23G7 


61 


463658 


785 


536342 


47 


14 


44G459 


723 


982331 


61 


464129 


784 


535871 


46 


15 


44G893 


722 


982294 


61 


464599 


783 


535401 


45 


If) 


447321) 


721 


982257 


61 


465069 


783 


534931 


44 


17 


447759 


720 


982220 


62 


465539 


782 


534461 


43 


18 


448191 


720 


982183 


62 


466008 


781 


533992 


12 


19 


448623 


719 


982146 


, 62 


466476 


780 


533524 


41 


20 


449054 


718 


982109 


62 


466945 


780 


533055 


40 


21 


9-449485 


717 


^9-982072 


62 


9-467413 


779 


.0-532587 


39 


22 


449915 


71G 


982035 


62 


467S80 


778 


532120 


38 


23 


450345 


716 


981998 


62 


468347 


778 


531653 


37 


24 


450775 


715 


981961 


62 


468814 


777 


531186 


36 


25 


451204 


714 


981924 


62 


469280 


77G 


530720 


35 


26 


451G32 


713 


981886 


62 


409746 


775 


530254 


34 


27 


452060 


713 


981849 


62 


470211 


775 


529789 


33 


28 


452488 


712 


981812 


62 


470676 


774 


529324 


32 


29 


452915 


711 


981774 


62 


471141 




528859 


31 


30 


453342 


710 


981737 


62 


471605 


773 


528395 


30 


31 


9-453768 


710 


9-981699 


G3 


9-472068 


772 


10-527932 


29 


32 


454194 


709 


981662 


63 


472532 


771 


527468 


28 


33 


454619 


708 


981625 


()3 


472995 


771 


527005 


27 


34 


455044 


707 


981587 


63 


473457 


770 


526543 


26 


35 


455469 


707 


98 J 549 


63 


473919 


769 


526081 


25 


36 


455893 


706 


981512 


63 


474381 


769 


525619 


24 


37 


456316 


7!)5 


981474 


63 


474842 


768 


525158 


23 


38 


45G739 


7U4 


981436 


63 


475303 


767 


524697 


oo 


39 


457162 


704 


981399 


63 


475763 


767 


524237 


21 


40 


457584 


703 


981 361 


63 


476223 


766 


523777 


20 


41 


9-458006 


702 


9-981323 


63 


9-476683 


765 


1 0-52331 7 


19 


42 


458427 


701 


981285 


63 


477142 


765 


522858 


]R 


43 


458848 


701 


981247 


63 


477601 


764 


522399 


17 


44 


45i»268 


700 


981209 


63 


478059 


7*53 


521941 


16 


45 


459688 


699 


981171 


63 


478517 


763 


521483 


J 5 


46 


460108 


698 


981133 


64 


478975 


7G2 


521025 


14 


47 


460527 


698 


981095 


64 


479432 


761 


5205ti8 


13 


48 


460946 


697 


981057 


64 


479889 


761 


520111 


12 


49 


4613G4 


696 


981019 


64 


480345 


760 


519655 


11 


50 


461782 


695 


980981 


64 


480801 


759 


519199 


10 


51 


9-462199 


C95 


9-i)80942 


64 


9-481257 


75!) 


10-518743 


9 


52 


462616 


694 


980904 


64 


481712 


758 


5 J 8288 


8 


53 


463032 


693 


l»808()6 


64 


482167 


757 


517833 


7 


54 


463448 


693 


980827 


64 


482621 


757 


517379 


6 


55 


463864 


6;)2 


980789 


()4 


483075 


756 


516925 


5 


56 


464279 


691 


980750 


64 


483529 


755 


516471 


4 


57 


464694 


690 


980712 


64 


483982 


755 


516018 


:( 


58 


465108 


690 


980673 


64 


484435 


754 


515565 


•2 


59 


465522 


689 


980635 


64 


484887 


753 


515113 


1 


60 


465935 


688 


980596 


64 


485339 


753 


514661 







{ Cosine 1 


( 


Sine 1 
72 


Degree 


Colang. 1 

S 




Tang. ' 


1A 



LOGARITHMIC SINES, COSINES, ETC. (17 Degrees.) 215 



M. / 



I D. I 



I D. 1 Tang 



I Colang. I 






f)-465935 


688 


9-98()596 


64 


9-485339 


755 


10-514661 


60 


J 


466348 


688 


980558 


64 


485791 


752 


514209 


59 


2 


466761 


687 


980519 


65 


486242 


751 


513758 


58 


3 


467173 


686 


980480 


65 


48(i693 


751 


513307 


57 


4 


467585 


085 


98(t442 


65 


487143 


750 


512857 


56 


5 


467996 


685 


980403 


65 


487593 


749 


512407 


55 


6 


468407 


684 


980364 


65 


488043 


749 


511957 


54 


7 


468817 


683 


980325 


65 


488402 


748 


511508 


53 


8 


469227 


683 


980286 


65 


488941 


747 


511059 


52 


9 


469637 


(i82 


980247 


65 


489390 


747 


510610 


51 


10 


470046 


681 


960208 


65 


489838 


746 


510162 


50 


]1 


9-470455 


680 


9-980169 


65 


9-4CC286 


746 


10-509714 


49 


12 


470863 


680 


980130 


65 


490733 


745 


509267 


48 


i:j 


471271 


679 


980091 


65 


491180 


744 


508820 


47 


]4 


471679 


678 


980052 


65 


491627 


744 


508373 


46 


J5 


472086 


078 


980012 


65 


492073 


743 


507927 


45 


16 


472492 


677 


979973 


65 


492519 


743 


507481 


44 


17 


472898 


67G 


979934 


66 


492965 


742 


507035 


43 


18 


473304 


676 


979895 


66 


493410 


741 


506590 


42 


19 


473710 


675 


979855 


66 


493854 


740 


506146 


41 


20 


474115 


674 


979816 


66 


494299 


740 


505701 


40 


21 


9-474519 


674 


9-979776 


66 


9-494743 


740 


10-505257 


39 


22 


474923 


(■)73 


979737 


66 


495186 


739 


504814 


38 


23 


475:i27 


072 


979697 


66 


495C30 


728 


504370 


37 


24 


475730 


672 


979658 


66 


496073 


737 


503927 


36 


25 


476133 


671 


979618 


66 


496515 


737 


503485 


35 


26 


47C53C 


670 


979579 


66 


496957 


736 


503043 


34 


27 


476938 


669 


979539 


66 


497399 


736 


502601 


33 


28 


477340 


669 


979499 


66 


497841 


735 


502159 


32 


29 


477741 


668 


979459 


66 


498282 


734 


501718 


31 


30 


478142 


667 


979420 


66 


498722 


734 


501278 


30 


31 


9-478542 


067 


9-979380 


66 


9-4C91G3 


733 


10-500837 


29 


32 


478942 


()G6 


979340 


66 


4G9CC3 


733 


500397 


28 


33 


479342 


G65 


979300 


67 


500042 


732 


499958 


27 


34 


479741 


6{i5 


979260 


67 


50C481 


731 


499519 


26 


35 


480140 


664 


979220 


67 


506920 


731 


499080 


25 


36 


480539 


663 


979180 


67 


601359 


730 


498641 


24 


37 


48(;<t37 


663 


979140 


67 


501797 


730 


498203 


23 


38 


48i:l34 


662 


979100 


67 


502235 


729 


497765 


22 


39 


481731 


661 


979059 


67 


5C2672 


728 


497328 


21 


40 


482128 


66 1 


979019 


67 


563109 


728 


496891 


20 


41 


9-482525 


660 


9-978979 


67 


9-503546 


727 


10-496454 


19 


42 


482921 


659 


978939 


67 


503982 


727 


496018 


18 


43 


483316 


659 


978898 


67 


504418 


726 


495582 


17 


44 


483712 


658 


978858 


67 


504854 


725 


495146 


16 


45 


484107 


657 


978817 


67 


505289 


725 


494711 


15 


46 


484501 


657 


978777 


67 


505724 


724 


494276 


14 


47 


484895 


656 


978736 


67 


506159 


724 


493841 


13 


48 


485289 


655 


978696 


68 


506593 


723 


493407 


12 


49 


4S5682 


655 


978655 


68 


507027 


722 


492973 


11 


50 


486075 


654 


978615 


68 


507460 


722 


492540 


10 


51 


9-486407 


653 


9-978574 


68 


9-507893 


721 


10-492107 


9 


52 


486860 


653 


978533 


68 


508326 


721 


491674 


8 


53 


487251 


652 


978493 


68 


508759 


720 


491241 


7 


54 


487643 


651 


978452 


68 


509191 


719 


490809 


6 


55 


488034 


651 


9784 11 


68 


509622 


719 


490378 


5 


56 


488424 


650 


978370 


68 


510054 


718 


489946 


4 


57 


488814 


650 


978329 


68 


510485 


718 


489515 


3 


58 


489204 


649 


978288 


68 


510916 


717 


489084 


2 


5£ 


489503 


648 


978247 


68 


511346 


716 


488654 


1 


60 


489982 


648 


978-206 


68 


511776 


716 


488224 






i Cowne I 



Cotang. ( 



I Tang. \ 



72 Degrees 



216 (18 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. 1 


• S,„e 


D. 1 


Cosine 


D. 1 


Tang. 1 


D. 


Cutang. 







9-489982 


648 


9-978206 


68 


9-511776 


716 


10-488224 


€0 


1 


490371 


648 


978165 


68 


512206 


716 


487794 


69 




490759 


647 


978124 


68 


512635 


715 


4873()5 


58 


fj 


491147 


646 


978083 


69 


513064 


714 


486936 


57 


4 


491535 


646 


978042 


69 


513493 


714 


486507 


50 


5 


491922 


645 


978001 


69 


513921 


713 


486079 


55 


(i 


492308 


644 


977959 


69 


514349 


713 


485651 


54 


7 


492695 


644 


977918 


69 


514777 


712 


485223 


53 


8 


493081 


643 


977877 


69 


515204 


712 


484796 


52 


9 


493466 


642 


977835 


69 


515631 


711 


484369 


51 


10 


493851 


G42 


977794 


69 


516057 


710 


483943 


50 


11 


9-494236 


C41 


9-977752 


69 


9-51 648A 


710 


10-483516 


49 


12 


494C21 


641 


977711 


69 


516910 


709 


483090 


48 


13 


495005 


G40 


977669 


69 


517335 


709 


482665 


47 


14 


495388 


639 


977628 


69 


517761 


708 


482239 


46 


15 


495772 


639 


977586 


69 


518185 


708 


481815 


45 


If) 


496154 


638 


977544 


70 


518610 


707 


481390 


44 


17 


496537 


637 


977503 


70 


519034 


706 


480966 


43 


18 


496919 


637 


977461 


70 


519458 


706 


480542 


43 


19 


497301 


636 


977419 


70 


519862 


705 


480118 


41 


20 


497682 


636 


977377 


70 


520305 


705 


479695 


40 


21 


9-498064 


035 


9-977335 


70 


9-520728 


704 


10-479272 


39 


22 


498444 


634 


977293 


70 


521151 


703 


478849 


38 


23 


498825 


634 


977251 


70 


521573 


703 


478427 


37 


24 


499204 


633 


977209 


70 


521995 


703 


478005 


36 


25 


499584 


632 


977167 


70 


522417 


702 


477583 


35 


2G 


499963 


632 


977125 


70 


522838 


702 


477162 


34 


27' 


500342 


631 


977083 


70 


523259 


701 


476741 


33 


28 


500721 


631 


977041 


70 


523080 


701 


476320 


32 


29 


501099 


630 


976999 


70 


524100 


700 


475900 


31 


30 


501476 


629 


970957 


70 


524520 


699 


475480 


30 


31 


9-501854 


029 


9-97r9J4 


70 


9-524939 


699 


10-475061 


29 


3-2 


502231 


628 


97G672 


71 


525359 


698 


474641 


28 


33 


502G07 


628 


97GS30 


71 


525778 


698 


474222 


27 


34 


502984 


627 


97G7S7 


71 


52G197 


697 


473803 


26 


35 


503360 


626 


976745 


71 


526015 


697 


473385 


25 


3G 


503735 


626 


970702 


71 


527033 


696 


472967 


24 


37 


504110 


025 


97G6G0 


71 


527451 


696 


472549 


23 


38 


504485 


6-:5 


97CG17 


71 


527808 


695 


472132 


22 


39 


504HG0 


024 


97C574 


71 


528285 


695 


471715 


21 


40 


505234 


623 


976532 


71 


528702 


694 


471298 


20 


41 


9-505608 


623 


9-970489 


71 


9-529119 


693 


10-470881 


19 


42 


505981 


622 


97(»446 


71 


529535 


693 


470465 


18 


43 


506354 


622 


976404 


71 


529950 


693 


470050 


17 


44 


506727 


621 


976361 


71 


530306 


692 


409634 


16 


45 


507099 


620 


97G318 


71 


530781 


691 


469219 


15 


4G 


507471 


020 


976275 


71 


531196 


691 


468804 


14 


47 


507843 


619 


97C232 


72 


531611 


690 


468389 


13 


48 


508214 


619 


970189 


72 


532025 


690 


4C7975 


12 


49 


508585 


618 


97G146 


72 


532439 


689 


467561 


11 


50 


508956 


618 


976103 


72 


532853 


689 


467147 


10 


51 


9-509326 


617 


9-97G0G0 


72 


9-533266 


688 


10-466734 


9 


52 


509G9G 


016 


97G017 


72 


533679 


688 


466321 


8 


53 


5100G5 


616 


975974 


72 


534092 


687 


465908 


7 


54 


510434 


615 


975930 


72 


534504 


687 


465496 


6 


55 


510803 


615 


975887 


72 


534916 


686 


405084 


5 


56 


511172 


614 


975844 


72 


535328 


086 


464672 


4 


57 


511540 


613 


975800 


72 


535739 


685 


404261 


3 


58 


511907 


613 


975757 


72 


530150 


685 


403850 


o 


59 


512275 


612 


975714 


72 


536561 


684 


463439 


1 


30 


512642 


612 


975670 


72 


536972 


684 


463028 






) Cosiiie \ 



I Sine I 



t Cotang-. t 



I Tang. 



7i Decrees. 



.{)<;. iRITI/MIC SIXES, COS/iVFS, ETC. (19 I'cgrees.) y]' 



M. I 


Sine 1 


D. I 


Cosine 1 


D. I 


Tans?. 1 


D. 1 


Co<angr. ( 







9 M2642 


612 


9-975670 


73 


9-53(iil72 


()84 


lt)-4630-28 


6(i 


1 


51 3009 


611 


975627 


73 


537382 


083 


4(n0l8 


59 


^ 


513375 


611 


975583 


73 


537792 


083 


40-'2u8 


58 


3 


513741 


610 


975539 


73 


538202 


682 


401798 


57 


4 


514107 


609 


975496 


73 


538611 


682 


461389 


50 


5 


514472 


609 


975452 


73 


539020 


681 


460980 


55 


6 


514837 


608 


975408 


73 


539429 


681 


400571 


54 


7 


515302 


608 


975-565 


73 


539837 


680 


400 J 03 


53 


8 


515566 


607 


9/5321 


73 


540245 


680 


459755 


52 


1) 


515930 


607 


975-277 


73 


540653 


679 


4593-17 


51 


10 


516294 


606 


975233 


73 


541061 


679 


458939 


50 


11 


9-510657 


605 


9-975189 


73 


9-541468 


678 


10-458532 


49 


12 


517020 


605 


975145 


73 


541875 


678 


4.TI-:5 


48 


13 


517382 


604 


975101 


73 


542-281 


677 


457719 


47 


14 


517745 


604 


975057 


73 


542388 


677 


4^:7312 


46 


15 


518107 


603 


975013 


73 


543094 


676 


45090() 


45 


16 


518468 


603 


9749G9 


74 


543439 


676 


450501 


44 


17 


518829 


602 


974925 


74 


543905 


675 


450095 


43 


18 


519190 


601 


974880 


74 


544310 


675 


455090 


42 


19 


519551 


601 


974836 


74 


544715 


674 


455285 


41 


20 


519911 


600 


974792 


74 


545119 


674 


454881 


40 


21 


9-52;)271 


600 


9-974748 


74 


9-545524 


673 


10-454476 


39 


22 


520631 


599 


974703 


74 


545J28 


673 


454072 


38 


23 


52U990 


599 


974G59 


74 


540331 


672 


4,:3009 


37 


24 


521349 


598 


974014 


74 


54GT35 


672 


453-265 


3(5 


25 


521707 


598 


974570 


74 


547138 


671 


452862 


35 


26 


5220G0 


597 


974525 


74 


547540 


071 


452400 


34 


27 


522424 


590 


974481 


74 


547943 


070 


45-2057 


33 


28 


522781 


5JG 


974436 


74 


548345 


070 


451655 


32 


29 


523138 


595 


974391 


74 


548747 


009 


451-253 


31 


30 


523495 


595 


974347 


75 


549149 


009 


450851 


30 


31 


9-523852 


594 


9-974302 


75 


9-549550 


608 


10-450450 


29 


32 


524208 


594 


974257 


75 


549JJ1 


6G8 


45;)049 


2.^ 


33 


524564 


593 


974212 


75 


55J3o2 


607 


449048 


27 


34 


524920 


593 


974167 


75 


550752 


667 


44J-248 


25 


35 


5-25275 


592 


9741-22 


75 


5511. ,2 


600 


448848 


25 


36 


525630 


591 


974077 


75 


55jj.^2 


606 


448448 


24 


37 


5-25984 


591 


974.^2 


75 


551952 


605 


448048 


23 


38 


526339 


590 


973J87 


75 


552351 


665 


447649 


22 


39 


526693 


590 


973942 


75 


552750 


665 


447250 


21 


40 


527046 


589 


973897 


75 


553149 


664 


440851 


2u 


41 


9-527400 


589 


9-973852 


75 


9-553548 


664 


10-446452 


19 


42 


627753 


588 


973807 


75 


553946 


663 


446054 


18 


43 


528105 


588 


973761 


75 


554344 


603 


445050 


17 


44 


528458 


587 


973716 


76 


554741 


6(52 


445259 


10 


45 


528810 


587 


973671 


76 


555139 


602 


444801 


15 


46 


52^161 


586 


9736-25 


7() 


555536 


601 


444404 


14 


• 47 


529513 


586 


973580 


76 


555933 


001 


444007 


13 


48 


529864 


585 


973535 


76 


55()32J 


609 


443071 


l-J 


4!) 


530215 


585 


973489 


76 


556725 


660 


443275 


!i 


50 


530565 


584 


9:3444 


76 


557121 


059 


44-2,^79 


1;« 


51 


9 530915 


584 


9-973398 


76 


9-557517 


659 


10-44-2483 


9 


52 


531265 


583 


973352 


76 


557913 


659 


442.)87 


H 


53 


531614 


582 


973307 


76 


558308 


658 


441092 


7 


54 


531963 


582 


973261 


76 


558702 


658 


441298 


n 


55 


532312 


§81 


973215 


76 


559097 


057 


440903 


5 


56 


532661 


581 


973169 


76 


559491 


657 


440509 


4 


57 


533009 


580 


973124 


76 


559885 


656 


4401 15 


3 


58 


533357 


580 


973078 


76 


560279 


656 


439721 


2 


59 


533704 


579 


973032 


77 


560673 


655 


439327 


1 


60 


534052 


578 


972986 


77 


56J066 


655 


438934 


1 



I Co«ne I 



irno I 



I Coung. 



I Tang. 



70 Degi-eea. 



218 (20 Degrees.) J.OGARfTTnrTC SINES, COSINES, ETC. 



I D, I Tang. | D. | Cotan?. ] 






9-534052 


578 


9-972986 


77 


9-561066 


655 


10-438034 


60 


1 


534399 


577 


972i)40 


77 


561459 


654 


438541 


59 


2 


534745 


577 


972894 


77 


561851 


654 


438149 


58 


3 


535092 


577 


972848 


77 


562244 


653 


437756 


57 


4 


535438 


576 


972802 


77 


5(52636 


653 


437364 


56 


5 


535783 


576 


972755 


77 


563028 


653 


436972 


55 


6 


536129 


575 


972709 


77 


563419 


652 


436581 


54 


7 


536474 


574 


972663 


77 


563811 


652 


436189 


53 


8 


536818 


574 


972617 


77 


5642„2 


651 


435798 


52 


9 


537163 


573 


97^2570 


77 


5(i4592 


651 


435408 


51 


10 


537507 


573 


972524 


77 


564983 


650 


435017 


50 


11 


9-537851 


572 


9-972478 


77 


9-565373 


650 


10-434627 


49 


12 


538194 


572 


972431 


78 


565763 


649 


434237 


48 


13 


538538 


571 


972385 


78 


566153 


649 


433847 


47 


14 


538880 


571 


972338 


78 


566542 


649 


433458 


46 


15 


539223 


570 


972291 


78 


566932 


648 


433068 


45 


16 


539565 


570 


972245 


78 


567320 


648 


432680 


44 


17 


539907 


569 


972198 


78 


567709 


647 


432291 


43 


18 


540249 


569 


972151 


78 


568098 


647 


431902 


42 


19 


540590 


568 


972105 


78 


568486 


646 


431514 


41 


20 


540931 


568 


972058 


78 


568873 


646 


431127 


4t' 


21 


9-541272 


567 


9-972011 


78 


9-569261 


645 


10-430739 


3& 


22 


541613 


567 


971964 


78 


569648 


645 


430352 


38 


23 


541953 


566 


971917 


78 


570035 


645 


429965 


37 


24 


542293 


566 


971870 


78 


570422 


644 


429578 


36 


25 


542632 


565 


971823 


78 


570809 


644 


429191 


35 


26 


542971 


565 


971776 


78 


571195 


643 


428805 


34 


27 


543310 


564 


971729 


79 


571581 


643 


428419 


33 


28 


543649 


564 


971682 


79 


571967 


642 


• 428033 


32 


29 


543987 


563 


971635 


79 


572352 


642 


427648 


31 


30 


544325 


563 


971588 


79 


572738 


642 


427262 


30 


31 


9-544663 


562 


9-971540 


79 


9-573123 


641 


10-426877 


29 


32 


545000 


562 


971493 


79 


573507 


641 


426493 


28 


33 


545338 


561 


971446 


79 


573892 


640 


426108 


27 


34 


545674 


561 


971398 


79 


574276 


640 


425724 


26 


35 


546011 


560 


971351 


79 


574660 


639 


425340 


25 


36 


548347 


560 


971303 


79 


575044 


639 


424956 


24 


37 


546683 


559 


971256 


79 


575427 


639 


424573 


23 


38 


547019 


559 


971208 


79 


575810 


638 


424190 


22 


39 


547354 


558 


971161 


79 


576193 


638 


423807 


21 


40 


547689 


558 


971113 


79 


576576 


637 


423424 


20 


41 


9-548024 


557 


9-971066 


80 


9-576958 


637 


10-423041 


19 


42 


548359 


557 


971018 


80 


577341 


636 


422659 


18 


43 


548093 


556 


970970 


8C 


577723 


636 


422277 


17 


44 


549027 


556 


970922 


80 


578104 


636 


421896 


16 


45 


549360 


555 


970874 


80 


578486 


635 


421514 


15 


46 


549693 


555 


970827 


80 


578867 


635 


421133 


14 


47 


550020 


554 


970779 


80 


579248 


634 


420752 


13 


48 


550359 


554 


970731 


80 


579629 


634 


420371 


12 


49 


550692 


553 


970683 


80 


580009 


634 


419991 


11 


50 


551024 


553 


970635 • 


80 


580389 


633 


419611 


10 


51 


4-551356 


552 


9-970586 


80 


9-580769 


633 


10-419231 


9 


52 


551687 


552 


970538 


80 


581149 


632 


418851 


8 


53 


552018 


552 


970490 


80 


581528 


632 


418472 


7 


54 


552349 


551 


970442 


80 


581907 


632 


418093 


6 


55 


552680 


551 


970394 


80 


582286 


631 


417714 


5 


56 


553010 


550 


970345 


81 


582665 


631 


417335 


4 


57 


553341 


550 


970297 


81 


583043 


G30 


41 6957 


3 


58 


553G70 


549 


970249 


81 


583422 


G30 


416578 


2 


59 


554000 


549 


970200 


81 


583800 


629 


416200 


1 


SO 


554329 


548 


970152 


81 


584177 


629 


415823 







Coaine 


1 


\ Sine 

69 


1 
Degre 


Cotang. 

es. 


1 


I Tang. 


1 M. 



LOGARITHMIC SmiCS, COSINES, ETC. [TL r)egrees.) 21ft 



M. I 



I Cosine | D. | Tang. 



I Cotang. I 



u 


9-554329 


548 


9-970152 


81 


9-584177 


629 


10-415823 


60 


1 


554658 


548 


970103 


81 


584555 


629 


415445 


59 


2 


554987 


547 


9T0055 


81 


584932 


628 


415068 


58 


3 


555315 


547 


97000G 


81 


585309 


628 


414691 


57 


4 


555643 


546 


969957 


81 


585686 


627 


414314 


56 


5 


555971 


546 


969909 


81 


586062 


627 


413938 


55 


6 


556299 


545 


969860 


81 


586439 


627 


413561 


54 


7 


556626 


545 


969811 


81 


58G815 


626 


413185 


53 


8 


556953 


544 


969762 


81 


587190 


G2G 


412810 


52 


9 


557280 


544 


969714 


81 


587566 


G25 


412434 


51 


10 


557606 


543 


969GG5 


81 


587941 


625 


412059 


50 


11 


9 557932 


543 


9-969616 


82 


9-58831G 


625 


10-411684 


49 


12 


558258 


543 


969567 


82 


588631 


624 


411309 


48 


13 


558583 


542 


969518 


82 


5890GG 


624 


410934 


47 


14 


558909 


542 


9G94G9 


82 


589440 


623 


410560 


46 


15 


559234 


541 


969420 


82 


589814 


623 


410186 


45 


16 


559558 


541 


969370 


82 


590188 


623 


409812 


44 


17 


559883 


540 


969321 


82 


590562 


622 


409438 


43 


18 


560207 


540 


969272 


82 


590935 


622 


409065 


42 


19 


500531 


539 


969223 


82 


591308 


622 


408692 


41 


20 


560855 


539 


969173 


82 


591681 


621 


4083] 9 


40 


21 


9-561178 


538 


9-969124 


82 


9-592054 


621 


10-407946 


39 


22 


561501 


538 


969075 


82 


592426 


620 


407574 


38 


23 


561824 


537 


969025 


82 


592798 


620 


407202 


37 


24 


562146 


537 


968976 


82 


593170 


619 


406829 


36 


25 


562468 


536 


968926 


83 


593542 


619 


4C6458 


35 


26 


562790 


536 


968877 


83 


593914 


618 


4CCC86 


34 


27 


563112 


536 


968827 


83 


594285 


618 


405715 


33 


28 


563433 


535 


968VV7 


83 


594656 


618 


405344 


32 


29 


563755 


535 


968728 


83 


595C27 


617 


404973 


31 


30 


564075 


534 


9C8678 


83 


5D5r.90 


617 


4046C2 


30 


31 


9-564396 


534 


9-968628 


83 


9-595768 


617 


10-404232 


29 


32 


5G4716 


533 


968578 


83 


59G138 


616 


403862 


28 


33 


565036 


533 


968528 


83 


59G508 


GIG 


403492 


27 


34 


565356 


532 


968479 


83 


596878 


GIG 


403122 


26 


35 


565676 


532 


968429 


83 


597247 


C15 


402753 


25 


36 


565995 


531 


968379 


83 


59761G 


615 


402384 


24 


37 


566314. 


531 


968329 


83 


597985 


615 


402015 


23 


38 


566632 


531 


968278 


83 


598354 


614 


401646 


22 


39 


566951 


530 


968228 


84 


598722 


614 


401278 


21 


40 


567269 


530 


968178 


84 


599091 


613 


400909 


20 


41 


9-567587 


529 


9-968128 


84 


9-599459 


613 


10-400541 


19 


42 


567904 


529 


968078 


84 


599827 


613 


400173 


18 


43 


568222 


528 


968027 


84 


600194 


612 


399806 


17 


44 


568539 


528 


967977 


84 


6C05G2 


612 


399438 


16 


45 


568856 


528 


967927 


84 


600929 


611 


299971 


15 


46 


569172 


527 


967876 


84 


601296 


611 


398704 


14 


47 


569488 


527 


9G782G 


.84 


C01662 


Oil 


398338 


13 


48 


569804 


526 


9G7775 


84 


602029 


610 


397971 


12 


49 


570120 


526 


967725 


84 


602395 


610 


397605 


11 


50 


570425 


525 


987G74 


84 


6027G1 


GIO 


397239 


10 


51 


9-570751 


525 


9-9G7G24 


84 


9-603127 


609 


10-396873 


9 


52 


5710GG 


524 


9G7573 


84 


603493 


609 


396507 


8 


53 


.':71S80 


594 


9G7522 


85 


603858 


eo9 


396142 


7 


54 


571G95 


523 


9G7471 


85 


604223 


608 


395777 


6 


55 


572009 


523 


967421 


85 


604588 


G08 


395412 


5 


56 


572323 


523 


967370 


85 


604953 


607 


395047 


4 


57 


572G3G 


522 


967319 


85 


605317 


607 


394683 


3 


58 


572950 


522 


967268 


85 


605682 


607 


394318 


2 


59 


573263 


521 


967217 


85 


C06046 


606 


393954 


1 


60 


573575 


521 


967166 


85 


606410 


606 


393590 






\ Cosine | 



I 

68 Degrees. 



Cotang. J 



I Tang. I M. 



220 (22 Derives.) LOGARITTIMTC, STNES, COSTXES, ETC, 



M. 


Sine 


1 D. 


Cosine 


1 D. 


1 Tan-. 


1 D. 


1 Cotan^. 




~0~ 


9-573575 


521 


9-967166 


85 


9-606410 


606 


10-393590 


~Gfl 


1 


573888 


520 


967115 


85 


606773 


606 


393227 


59 


2 


574200 


520 


967064 


85 


607137 


605 


392863 


58 


3 


574512 


519 


9G7013 


85 


607500 


605 


392500 


57 


4 


574824 


519 


9GG9G1 


85 


607863 


604 


392137 


56 


5 


575136 


519 


96G910 


85 


,608-225 


604 


391775 


55 


6 


575447 


518 


9GG859 


85 


608588 


604 


391412 


54 


7 


575758 


518 


966808 


85 


608950 


603 


391050 


53 


8 


576069 


517 


98675G 


86 


609312 


603 


390G83 


52 


9 


57G379 


517 


966705 


86 


609674 


603 


390326 


51 


10 


576689 


516 


966653 


86 


610036 


602 


389964 


50 


Jl 


9-576999 


516 


9-966602 


86 


9-61U39V 


602 


10-389603 


49 


12 


577309 


516 


9GG550 


86 


610759 


602 


389241 


48 


13 


577G18 


515 


966499 


86 


611120 


GOl 


388880 


47 


14 


577927 


515 


9GG447 


86 


611480 


601 


388520 


46 


15 


578236 


514 


96G395 


86 


611841 


601 


388159 


45 


IG 


578545 


514 


966344 


86 


612201 


600 


387799 


44 


17 


578853 


513 


9G6292 


86 


612561 


600 


387439 


43 


18 


579162 


513 


966240 


86 


612921 


600 


387079 


42 


19 


579470 


513 


966188 


86 


613281 


599 


386719 


41 


20 


579777 


512 


966136 


86 


613641 


599 


386359 


40 


21 


9-580085 


512 


9 966085 


87 


9-614000 


598 


10-386000 


39 


oo 


580392 


511 


966033 


87 


614359 


598 


385641 


38 


23 


580699 


511 


965981 


87 


614718 


598 


385282 


37 


24 


581005 


511 


965928 


87 


615077 


597 


384923 


36 


25 


581312 


510 


9G5876 


87 


615435 


597 


384565 


35 


26 


581618 


510 


965824 


87 


615793 


597 


384207 


34 


27 


581924 


509 


965772 


87 


616151 


596 


383849 


33 


2H 


5S-2229 


509 


965720 


87 


61G509 


596 


383491 


32 


2.:) 


5^^2535 


509 


965668 


87 


61G867 


596 


383133 


31 


3.1 


582840 


508 


965615 


87 


617224 


595 


382770 


30 


31 


y-583145 


508 


9-9655G3 


87 


9-617582 


595 


10-382418 


29 


32 


583449 


507 


9G5511 


87 


617939 


595 


382061 


28 


33 


583754 


507 


965458 


87 


6182[»5 


594 


381705 


27 


34 


584058 


506 


965406 


87 


618652 


594 


381348 


26 


35 


584361 


506 


9G5353 


88 


619008 


594 


380992 


25 


36 


584665 


506 


965301 


88 


619364 


593 


380636 


24 


37 


5849G8 


505 


965248 


88 


619721 


593 


380279 


23 


38 


585272 


505 


965195 


88 


620076 


593 


379924 


22 


39 


5;^5574 


504 


965143 


88 ■ 


620432 


592 


379568 


21 


40 


5;i5S77 


504 


9G5090 


88 


G20787 


592 


379213 


20 


41 


9-5''6l79 


503 


9-9G5037 


88 


9-621142 


592 


10-378858 


19 


42 


586482 


503 


9G4984 


88 


621497 


591 


378503 


18 


43 


5?6783 


503 


9G4931 


88 


621852 


591 


378148 


17 


44 


5S7085 


502 


964879 


88 


G22207 


590 


377793 


10 


45 


5S7386 


502 


964826 


88 


622561 


590 


377439 


15 


40 


5^7688 


501 


964773 


88 


622915 


590 


377085 


14 


47 


587989 


501 


964719 


88 


623269 


589 


376731 


13 


48 


588289 


501 


9G4GG6 


89 ■ 


623623 


589 


376377 


12 


49 


588590 


500 


9G4613 


89 


623976 


589 


37C024 


11 


50 


588890 


500 


964560 


89 


624330 


588 


375670 


10 


5! 


9-58yi90 


499 


9-964507 


89 


9-624683 


588 


10-375317 


9 


52 


5^^9489 


499 


964454 


89 


625036 


588 


374964 


8 


53 


5^9789 


409 


964400 


89 


625388 


587 


374612 


7 


54 


590088 


498 


964347 


89 


625741 


587 


374259 


6 


55 


590387 


498 


964294 


89 


626093 


587 


373907 


5 


56 


590086 


497 


9f)4240 


89 


626445 


586 


373555 


4 


57 


590984 


497 


9G4187 


89 


G26797 


580 


373203 


3 


58 


591282 


497 


964133 


89 


627149 


580 


372851 


2 


59 


591580 


496 


964080 


89 


627501 


585 


372499 


1 


60 


591878 


496 


964026 


89 


627852 


585 


372148 






i Cosine I 



Sine I I Cotang'. J 

67 Degrees. 



) Tang. 



LOGAUITIOIIC SnVES, COS /XL'S, ETC. (2:i Dviiwcf^.) 221 



M. 


Sine 


u. 1 


Cosine 


D. 


Tan?. 


D. 


Comng. 







~9-59T878^ 


496 


9-964026 


89 


9-627852 


585 


10-372148 


60 


1 


592176 


495 


963972 


89 


628203 


585 


371797 


59 


2 


592473 


495 


963919 


89 


6285.54 


585 


371446 


58 


3 


592770 


495 


963865 


90 


628905 


584 


371095 


57 


4 


593067 


494 


963811 


90 


629255 


584 


370745 


56 


5 


593363 


494 


963757 


90 


629600 


583 


370394 


55 


6 


593659 


493 


963704 


90 


()29956 


583 


370044 


54 


7 


593955 


493 


963650 


90 


(>30306 


583 


369694 


53 


8 


594251 


493 


963596 


90 


()30656 


583 


369344 


52 


<) 


594547 


492 


963542 


90 


631005 


582 


368995 


51 


i!; 


594842 


492 


96;i488 


90 


631355 


582 


368645 


50 


Ji 


9-595137 


491 


9-90:i434 


90 


9-631704 


582 


10-368296 


49 


1-2 


595432 


491 


963379 


90 


632053 


581 


3r.7947 


48 


iA 


595727 


491 


963325 


90 


632401 


581 


3(i7599 


47 


14 


596021 


490 


903271 


90 


632750 


581 


367250 


46 


15 


596315 


490 


963217 


90 


633098 


580 


366902 


45 


iK 


596C09 


489 


963163 


90 


633447 


580 


366553 


44 


17 


596903 


489 


963108 


91 


633795 


580 


366205 


43 


18 


597196 


489 


963054 


91 


634143 


579 


365857 


42 


19 


597490 


488 


962999 


91 


634490 


579 


365510 


41 


20 


597783 


488 


962945 


91 


634838 


579 


365162 


40 


21 


9-59P075 


487 


9-962890 


91 


9-635185 


578 


10-364815 


39 


22 


598368 


487 


962836 


91 


635532 


578 


364468 


38 


23 


598660 


487 


962781 


91 


635879 


578 


364121 


37 


24 


598952 


486 


962727 


91 


636226 


577 


363774 


36 


25 


599244 


486 


962672 


91 


636572 


577 


363428 


35 


2C 


599536 


485 


962617 


91 


636919 


577 


363081 


34 


27 


599827 


485 


962562 


91 


637265 


577 


362735 


33 


28 


600118 


485 


9C2508 


91 


637611 


576 


362389 


32 


29 


600409 


484 


962453 


91 


637956 


576 


362044 


31 


30 


600700 


484 


962398 


92 


638302 


576 


361698 


30 


31 


9-000990 


484 


9-962343 


92 


9-638647 


575 


10-361353 


29 


32 


601280 


483 


962288 


92 


638992 


575 


361008 


28 


33 


601570 


483 


962233 


92 


639337 


575 


360663 


27 


34 


691860 


482 


962178 


92 


639682 


574 


360318 


26 


35 


602 I5U 


482 


962123 


92 


640027 


574 


359973 


25 


36 


602439 


482 


962067 


92 


640371 


574 


359629 


24 


37 


602728 


481 


962012 


92 


640716 


573 


359284 


23 


38 


603017 


481 


961957 


92 


641060 


573 


358940 


22 


39 


603305 


481 


961902 


92 


641404 


573 


358596 


21 


40 


603594 


480 


961846 


92 


641747 


572 


358253 


20 


41 


9-603882 


480 


9-961791 


92 


9-642091 


572 


10 357909 


19 


42 


6G4170 


479 


961735 


92 


642434 


572 


357566 


18 


43 


604457 


479 


961680 


92 


642777 


572 


357223 


17 


44 


604745 


479 


901624 


93 


643120 


571 


356880 


16 


45 


605032 


478 


961569 


93 


643463 


571 


356537 


15 


4G 


005319 


478 


961513 


93 


643806 


571 


356194 


14 


47 


605606 


478 


961458 


93 


644148 


570 


355852 


13 


48 


605892 


477 


961402 


93 


644490 


570 


355510 


12 


49 


606 J 79 


4 ;~ 


961340 


93 


644832 


570 


355168 


11 


50 


606465 


476 


961290 


93 


645174 


569 


354826 


10 


51 


9-606751 


476 


9-961235 


93 


9-645516 


569 


10-354484 


9 


52 


607036 


476 


961179 


93 


645857 


569 


354143 


8 


53 


607322 


475 


961123 


93 


646199 


5(39 


353801 


7 


54 


607607 


475 


961067 


93 


646540 


568 


353460 


6 


55 


607892 


474 


961011 


93 


646881 


568 


353119 


5 


56 


608177 


474 


960955 


93 


647222 


568 


352778 


4 


57 


608461 


474 


960899 


93 


647562. 


567 


352438 


3 


58 


608745 


473 


960843 


94 


647903 


567 


352097 


2 


59 


609029 


473 


960786 


94 


648243 


567 


351757 


1 


60 


609313 


473 


960730 


94 


648583 


566 


351417 






Coane | 



Sine I 



Cotang-. I 



Tang. I Bl 



66 Degrees. 



222 (24 I)i-giec's.) LOGARITHMIC SIXES, COSINES, ETC. 



Sine 

UW>93]y 
J3()yo97 
609f<80 
610IH4 
610447 
6 J 0729 
fi]J(ll2 
61J:i94 
611576 
61lri58 
612140 

9-612421 
612-1(2 
612983 
6] 3264 
6 J 3545 
613825 
61411)5 
6)4385 
6I4H65 
614944 

9-615223 
615502 
615781 



616338 



617172 
617450 
617727 

9-618004 
618281 
618558 
618834 
619110 
619386 
619662 
619938 
6202 J 3 
620488 

9-620763 
621038 
62)3)3 
621587 
621861 
622135 
622409 
622682 
622!»56 
623229 

9-623502 
623774 
624047 
6243)9 
62459) 
624863 
625 J 35 
62540H 
625677 



D. 


Cosine | 


D. 1 


Tang. 1 


D. 


Cotang. 




473 


9-960730 


94 


9-648583 


566 


10 351417 


"60 


472 


960674 


94 


648923 


566 


351077 


5S 


472 


960618 


94 


649263 


566 


350737 


58 


472 


960.itil 


94 


649602 


566 


350398 


57 


471 


960505 


94 


649942 


565 


350058 


56 


471 


960448 


94 


6.50281 


565 


349719 


5i 


40 


960392 


94 


6.50620 


5hi5 


349380 


54 


4 1) 


96o:«5 


94 


650959 


564 


349041 


53 


4 ) 


960279 


94 


651-297 


51 ;4 


348703 


52 


4'k) 


960-222 


94 


651636 


564 


348364 


5 


4()9 


960 i 65 


94 


651974 


563 


348026 


5t 


4r>9 


9-960109 


95 


9-6.5-2312 


563 


10-347688 


4L 


4nrt 


900002 


95 


65-2650 


563 


347350 


48 


4r>8 


959995 


95 


652988 


563 


3470)2 


4" 


4h7 


959938 


95 


653326 


562 


34t)674 


4t 


4ti7 


959882 


95 


653663 


562 


346337 


4i 


4h7 


959825 


95 


654000 


562 


346000 


4-f 


4(i6 


959768 


95 


654337 


561 


345663 


4: 


466 


9597 1 1 


95 


654674 


561 


345326 


4; 


466 


959654 


95 


6.V)0ll 


561 


344989 


4 


465 


959596 


95 


655348 


561 


344652 


4( 


465 


9-959539 


95 


9-655684 


560 


10-344316 


3i 


465 


959482 


95 


656020 


560 


343980 


3t 


464 


959425 


95 


656356 


560 


343644 


3- 


464 


959368 


95 


656692 


559 


343308 


3t 


464 


959310 


9t; 


657028 


559 


342972 


3. 


463 


959253 


96 


657364 


559 


342636 


3^ 


463 


959195 


96 


657699 


559 


342301 


3: 


462 


959138 


96 


658034 


558 


341966 


3; 


462 


959081 


96 


658369 


558 


341631 


3 


462 


959023 


96 


658704 


558 


341296 


3( 


461 


9-958965 


96 


9-659039 


558 


10-340961 


2i 


461 


958908 


WJ 


659373 


5.57 


340627 


2i 


461 


958850 


96 


659708 


557 


340292 


2- 


460 


958792 


96 


660042 


557 


339958 


2( 


460 


958734 


96 


660376 


557 


339624 


2. 


460 


958677 


96 


660710 


556 


339290 


2^ 


459 


958619 


96 


661043 


556 


338957 


2: 


459 


958561 


96 


66 J 377 


556 


338623 


2S 


459 


958503 


97 


661710 


555 


338290 


2 


458 


958445 


97 


662043 


555 


337957 


2( 


458 


9-958387 


97 


9-662376 


5.55 


10-337624 




457 


958329 


97 


662709 


554 


337291 




457 


9.58271 


97 


663042 


554 


336958 




457 


958213 


97 


663375 


5.54 


336625 




456 


958)54 


97 


663707 


554 


336293 




456 


95H096 


97 


6640.S9 


553 


335961 




456 


958038 


97 


664371 


553 


335629 




455 


957979 


97 


664703 


5.)3 


335297 




455 


957921 


97 


66.5035 


553 


334965 




455 


957863 


97 


665366 


552 


334634 




454 


9-957804 


97 


9-66.5697 


5.52 


10-334.303 




454 


957746 


98 


666029 


552 


333S7] 




a:a 


957687 


98 


666360 


551 


333640 




453 


957628 


98 


666691 


551 


333309 




453 


957570 


98 


667021 


551 


332979 




453 


9575)1 


98 


6673.'»2 


551 


332648 




452 


957452 


98 


667(i82 


550 


332318 




452 


957393 


98 


668013 


550 


331987 




452 


957335 


98 


668343 


550 


3316.57 




451 


957276 


98 


668672 


550 


331328 





1 Cosine I 



i Sme 



I Coiang. I 



Tang. 



65 Degrees. 



KUiARITTTMIC STXES, COSIKES, ETC. (2:. Doki-pps.) 22S 



Sine I 



I Cosine | .D. | Tang. | 






9-625948 


451 


9-957276 


98 


9-668673 


550 


|0-3:(I327 


60 


J 


526219 


451 


957217 


98 


669002 


549 


330998 


59 


2 


626490 


451 


957158 


98 


669332 


549 


330668 


58 


3 


626760 


450 


957099 


98 


669661 


549 


330339 


57 


4 


627030 


450 


957040 


98 


669991 


548 


330009 


.56 


5 


6273^)0 


450 


956981 


98 


670320 


548' 


329680 


55 


6 


6275 70 


449 


956921 


99 


670649 


548 


329351 


54 


7 


627M0 


449 


956862 


99 


670977 


548 


329023 


53 


8 


628 J 09 


449 


956803 


99 


671306 


547 


328694 


.52 


9 


628378 


448 


956744 


99 


671634 


547 


328366 


.51 


10 


628647 


448 


956684 


99 


671963 


547 


328037 


50 


11 


"> 628916 


447 


9-956625 


99 


9-672291 


547 


10-327709 


49 


12 


629 H5 


447 


956566 


99 


' 672619 


546 


327X81 


48 


13 


629453 


447 


956506 


99 


672947 


546 


327(153 


47 


14 


62^721 


446 


956447 


99 


673274 


546 


326726 


46 


15 


629989 


446 


956387 


99 


673602 


546 


326:^98 


45 


16 


630257 


446 


95H327 


99 


673929 


545 


326(171 


■^4 


17 


63II.V24 


446 


956263 


99 


674257 


545 


325- 43 


A 1 


iA 


SHU-; 92 


445 


956208 


100 


M', 4584 


545 


325JI6 


4 . 


J9 


6311(59 


445 


956148 


J 00 


674910 


544 


3251 "HO 


41 


20 


63 J 326 


445 


956089 


J 00 


675237 


544 


324163 


40 


21 


9-631593 


444 


9-956029 


100 


9-675564 


544 


10-324436 


39 


22 


631859 


444 


955969 


100 


675890 


544 


324110 


3H 


23 


632125 


444 


955909 


100 


676216 


543 


323784 


37 


24 


632392 


443 


955849 


100 


676543 


543 


323457 


36 


2o 


6326.58 


443 


955789 


100 


676869 


543 


323131 


35 


2« 


R32923 


443 


955729 


100 


677194 


543 


322806 


34 


27 


633189 


442 


955669 


100 


677520 


542 


322480 


33 


28 


633454 


442 


955609 


100 


677846 


542 


3221.54 


32 


2!» 


533719 


442 


955548 


100 


678171 


542 


32IH29 


31 


30 


633984 


441 


955488 


100 


678496 


542 


321504 


30 


31 


9-634249 


441 


9-955428 


101 


9-678821 


541 


10-321179 


29 


32 


6:}4514 


440 


955368 


101 


679146 


541 


32(»8.54 


28 


33 


634778 


440 


955307 


101 


679471 


541 


320529 


27 


34 


635042 


440 


955247 


101 


679795 


541 


3202(t5 


26 


35 


6:;5306 


439 


955186 


101 


680120 


540 


319880 


25 


36 


635570 


439 


9551-26 


101 


680444 


540 


319.5.56 


24 


37 


fi35H34 


439 


955065 


101 


680768 


540 


319232 


23 


38 


636097 


438 


955005 


101 


681092 


540 


318908 


22 


39 


636360 


438 


954944 


101 


681416 


539 


318.584 


21 


40 


636623 


438 


954883 


101 


681740 


539 


318260 


20 


41 


9-636886 


437 


9-954823 


101 


9-682063 


539 


10-3179.37 


19 


42 


637148 


437 


954762 


101 


682387 


539 


317613 


\S 


43 


637411 


437 


954701 


101 


682710 


538 


317290 


17 


44 


637673 


437 


954640 


101 


683033 


538 


316967 


16 


45 


637935 


436 


954579 


101 


683356 


538 


316644 


15 


46 


638197 


436 


954518 


102 


683679 


538 


316321 


14 


47 


638458 


436 


954457 


102 


684001 


537 


31.5999 


13 


48 


6387-20 


435 


954396 


102 


684324 


537 


315676 


12 


49 


638981 


435 


954335 


102 


684646 


537 


3153.54 


11 


50 


639242 


435 


954274 


102 


684968 


537 


315032 


10 


51 


9-630503 


434 


9-954213 


102 


9-685290 


536 


10-314710 


9 


52 


639764 


434 


954152 


102 


685612 


536 


314388 


8 


53 


640024 


434 


954090 


102 


6859:M 


536 


314066 


7 


54 


640284 


433 


954(t29 


1(12 


686255 


536 


31.3745 


6 


55 


640544 


433 


9539«i8 


102 


686577 


535 


313423 


5 


5fi 


640804 


433 


953906 


102 


686898 


535 


313102 


4 


57 


64illt)4 


432 


953845 


102 


687219 


535 


312781 


3 


58 


641324 


432 


953783 


102 


687540 


535 


312460 


2 


59 


641584 


432 


953722 


103 


687861 


534 


312139 


1 


60 


641842 


431 


953660 


103 


688182 


534 


311818 






\ Sine I 



I Cotaiig. I 



I TAag. J U. 



224 


(20 Deg 


rces.) LOGARITHMIC 


SIA'^FS, COS/^'FS, ETC. 




M. 


Sine 


, D. 


Cosine 


D. 


Tang. 


D. 1 


Cotang. 


1 





9-641842 


431 


9-953660 


103 


9-688182 


534 


10-311818 


60 




6 1.21 01 


431 


953599 


103 


688502 


534 


311498 


59 


2 


w^^3r*o 


421 


953537 


103 


088823 


534 


311177 


58 


3 


642618 


430 


953475 


103 


689143 


533 


310857 


57 


4 


642877 


430 


953413 


103 


•689463 


533 


310537 


56 


5 


643135 


430 


953352 


103 


689783 


533 


310217 


55 


6 


643393 


430 


953290 


103 


690103 


533 


309897 


54 


7 


64365'-) 


429 


953228 


103 


690423 


533 


309577 


53 


8 


6439U8 


429 


9531CG 


103 


690742 


532 


309258 


52 


9 


644105 


429 


953104 


103 


691 002 


532 


308938 


51 


10 


6444':3 


428 


953042 


1C3 


691281 


532 


308GI9 


50 


11 


9 644C80 


428 


9-952980 


104 


9-691700 


531 


10-308300 


49 


12 


64493G 


428 


952918 


104 


092019 


531 


307i)81 


48 


13 


64519;] 


427 


952855 


104 


092338 


531 


307002 


47 


14 


645450 


427 


952793 


104 


092056 


531 


307344 


46 


15 


64570G 


427 


952731 


104 


692975 


531 


307025 


45 


16 


6439C2 


426 


952G69 


104 


693293 


530 


300707 


44 


17 


64G218 


426 


9526C6 


104 


693012 


530 


30G388 


43 


18 


646474 


426 


952544 


104 


093930 


530 


306070 


42 


19 


646729 


425 


952481 


104 


G94248 


530 


305752 


41 


20 


64G984 


425 


952419 


104 


094566 


529 


305434 


40 


21 


9-G47240 


425 


o-o:;2356 


104 


9-C94883 


529 


10-305117 


39 


22 


647494 


424 


952294 


104. 


695201 


529 


304799 


38 


23 


647749 


424 


952231 


104 


695518 


529 


304482 


37 


24 


C480G4 


4C4 


9521C8 


105 


G05C3G 


529 


304164 


36 


25 


648258 


424 


9521C6 


105 


CDG153 


528 


303847 


35 


26 


648512 


423 


952043 


105 


030470 


528 


303530 


34 


27 


6487CG 


423 


951980 


105 


69G787 


528 


303213 


33 


28 


649020 


423 


951917 


105 


697103 


528 


302897 


32 


29 


649274 


422 


951854 


105 


697420 


527 


302580 


31 


30 


649527 


422 


(,51791 


105 


69773G 


527 


3022G4 


30 


31 


9-649781 


422 


<;-951728 


1C5 


9-698053 


527 


10-301947 


29 


32 


650034 


•i' i 


951GG5 


105 


698309 


527 


301031 


28 


33 


650287 


421 


951G02 


105 


698085 


526 


301315 


07 


34 


650539 


421 


95]5;:9 


105 


699001 


520 


300999 


26 


35 


C50792 


421 


951476 


105 


6993 IG 


526 


300684 


25 


36 


G51044 


420 


951412 


105 


699032 


526 


300368 


24 


37 


651297 


420 


951349 


106 


099947 


526 


300053 


23 


38 


651549 


420 


951286 


100 


700203 


525 


299737 


22 


39 


G51800 


419 


951222 


106 


700578 


525 


299422 


21 


40 


652052 


419 


951159 


106 


700893 


525 


299107 


20 


41 


9-652304 


419 


9-951096 


106 


9-701208 


524 


10-298792 


19 


42 


652555 


418 


951032 


106 


701523 


524 


298477 


18 


43 


G52806 


418 


950968 


106 


701837 


524 


298163 


17 


44 


653057 


418 


950905 


106 


702152 


524 


297848 


16 


45 


653308 


418 


950841 


106 


702406 


524 


297534 


15 


46 


053558 


417 


950778 


lOG 


702780 


523 


297220 


14 


47 


653808 


417 


950714 


106 


7G3U95 


523 


296905 


13 


48 


654059 


4.7 


950650 


106 


703409 


523 


29()591 


12 


49 


654309 


410 


950586 


106 


703723 


523 


290277 


11 


50 


654558 


416 


950522 


107 


704036 


522 


295964 


10 


51 


654808 


416 


9-950458 


107 


9-704350 


522 


10-295650 


9 


52 


655058 


416 


950394 


107 


704()63 


522 


295337 


S 


53 


655307 


415 


950330 


107 


704977 


522 


295023 


7 


54 


(J55."5G 


415 


950266 


107 


705290 


522 


294710 


6 


55 


6558(15 


415 


950202 


107 


705603 


521 


294397 


5 


56 


65(5054 


414 


950138 


107 


705910 


521 


294084 


4 


57 


656302 


414 


950074 


107 


706228 


521 


293772 


3 


58 


656551 


414 


950010 


107 


706541 


521 


293459 


it 


59 


656799 


413 


949945 


107 


70fi854 


521 


293146 


1 


60 


657047 


413 


949881 


1(»7 


707166 


520 


292834 







1 C«une 


1 


1 Sine 

63 


1 
Degre 


Cotang. 




Ta..g. 1 


M. 



LOGARirmilC SIXES, COSTXES, ETC. (27 Degrees.) 225 



M. 


Sine 


D. 


1 Cosine 


1 D. 


Tang. 


D. 


Coiaiig. 


1 





9-657047 


413 


9-949881 


107 


9-707166 


520 


10-292834 


60 


1 


657295 


413 


949816 


107 


707478 


520 


292522 


59 


2 


657542 


412 


94y7o2 


107 


707790 


5ii0 


292210 


58 


3 


657790 


412 


949688 


108 


708102 


520 


291898 


57 


4 


658037 


412 


949623 


108 


708414 


519 


291586 


56 


5 


658284 


412 


949558 


108 


708726 


519 


291274 


55 


6 


658531 


411 


949494 


108 


709037 


519 


290963 


54 


7 


658778 


411 


949429 


108 


709349 


519 


290651 


53 


8 


659025 


411 


94t»364 


108 


709660 


519 


290340 


52 


!> 


659271 


410 


949300 


108 


709971 


518 


290029 


51 


](• 


659517 


4J0 


949235 


108 


710282 


518 


289718 


50 


11 


9-659763 


410 


9-949170 


108 


9-710593 


518 


10-289407 


49 


12 


66U009 


409 


949 1 05 


108 


710904 


5,8 


289096 


48 


13 


660255 


4(t9 


949040 


108 


711215 


518 


288785 


47 


14 


6605(11 


409 


948975 


108 


711525 


5J7 


288475 


46 


15 


630746 


409 


948910 


108 


711836 


517 


288164 


45 


lo 


6609<»1 


408 


948845 


108 


712148 


517 


28V854 


44 


17 


661236 


408 


948780 


109 


712456 


517 


287544 


43 


]• 


661481 


4J8 


948715 


109 


712766 


516 


287234 


42 


19 


661726 


407 


948650 


109 


713076 


516 


286924 


41 


2U 


661970 


407 


948584 


109 


713386 


516 


286614 


40 


21 


9-662214 


407 


9-948519 


109 


9-713696 


516 


1(1-286304 


39 


22 


662459 


407 


948454 


109 


714005 


5l() 


285995 


38 


23 


662703 


406 


948388 


109 


714314 


515 


285686 


37 


24 


662946 


406 


948323 


109 


714624 


515 


. 285376 


36 


25 


683190 


406 


948257 


109 


714933 


515 


285067 


35 


26 


663433 


405 


948192 


109 


715242 


515 


284758 


34 


27 


663677 


405 


948126 


109 


715551 


511 


284449 


33 


28 


663920 


405 


948060 


109 


715S60 


514 


284140 


32 


29 


664163 


405 


947995 


110 


716168 


514 


283832 


31 


30 


664406 


404 


941929 


110 


716477 


514 


983523 


30 


31 


9-664648 


404 


9-947863 


110 


9-716785 


514 


](l-283215 


29 


32 


664891 


4(14 


947797 


110 


717093 


513 


282907 


28 


33 


665133 


403 


947731 


110 


717401 


513 


282599 


27 


34 


665375 


403 


947665 


110 


717709 


513 


282291 


26 


35 


665617 


403 


947600 


110 


718017 


513 


281983 


25 


36 


665859 


402 


947533 


110 


718325 


513 


281675 


24 


37 


666100 


402 


947467 


110 


718633 


512 


281367 


23 


38 


666342 


402 


947401 


110 


718940 


512 


281060 


22 


39 


666583 


402 


947335 


110 


719248 


512 


280752 


21 


40 


666824 


401 


947269 


110 


719555 


512 


280445 


20 


41 


9-367065 


401 


9-94/203 


110 


9-719862 


512 


10-280138 


19 


42 


667305 


401 


947136 


111 


720169 


511 


279831 


18 


43 


667546 


401 


947070 


111 


720476 


511 


279524 


17 


44 


667786 


400 


947004 


111 


720783 


51i 


279217 


16 


45 


668027 


400 


946937 


111 


721089 


511 


278911 


15 


46 


668267 


400 


946871 


111 


721396 


511 


278604 


14 


47 


668506 


399 


946804 


111 


721702 


510 


278298 


13 


48 


668746 


399 


946738 


111 


722009 


510 


277991 


12 


49 


668986 


399 


946671 


111 


722315 


510 


277685 


11 


50 


669225 


399 


946604 


111 


722621 


510 


277379 


10 


51 


9-669464 


398 


9-946538 


111 


9-722927 


510 


10277073 


9 


52 


669703 


398 


946471 


111 


723232 


509 


2767(58 


8 


53 


669942 


398 


946404 


111 


723538 


509 


276462 


7 


54 


670 181 


397 


946337 


111 


723844 


509 


276156 


6 


55 


670419 


397 


946270 


112 


724149 


509 


275851 


5 


56 


670658 


397 


946203 


112 


724454 


509 


275546 


4 


57 


67(1896 


397 


946136 


112 


724759 


508 


275241 


3 


58 


671134 


396 


946069 


112 


725065 


508 


274935 


2 


50 


671372 


396 


9460(12 


112 


725369 


508 


274631 


1 


6U 


671609 


396 


945935 


112 


725674 


508 


274326 






I Coeine | 



I I Coiang, I 

62 Degrees. 



Tanj. I M 



226 (28 Degrees.) LOOAHITHMTC SINES, COSINES, ETC. 



I Cosine | 



I Tang. 



I Cotang. I 



Cofine 



I I Cotang. 

^ Pegr«eii. 






9-671609 


396 


9-945935 


112 


9-725K74 


508 


10-274326 


60 


1 


671847 


395 


345868 


l.i 


7-.i597y 


508 


274021 


59 


2 


672084 


395 


945800 


li2 


726284 


507 


273716 


58 


3 


672321 


395 


945733 


112 


726588 


507 


273412 


57 


4 


672558 


395 


945666 


112 


726892 


507 


273108 


56 


5 


672795 


394 


945598 


112 


727197 


507 


272803 


55 


6 


673032 


394 


945531 


112 


727501 


507 


272499 


54 


■ 7 


673268 


394 


945464 


113 


727805 


506 


272195 


53 


8 


673505 


394 


945396 


113 


728109 


.506 


27l«91 


52 


9 


673741 


393 


945328 


113 


728412 


.506 


271,588 


51 


10 


673977 


393 


945261 


113 


728716 


506 


271284 


511 


11 


9-674213 


393 


9-945193 


113 


9-729020 


506 


10-270980 


49 


12 


674448 


392 


945125 


113 


7293-23 


505 


270677 


48 


13 


6746H4 


392 


945058 


113 


729626 


505 


270374 


47 


14 


674919 


392 


944990 


113 


729929 


505 


270071 


46 


15 


675155 


392 


944922 


113 


730233 


505 


269767 


45 


16 


675390 


391 


944854 


113 


730535 


505 


269465 


44 


17 


675624 


391 


944786 


113 


730838 


504 


269162 


43 


18 


675859 


391 


944718 


113 


731141 


504 


26rtH59 


42 


39 


676094 


391 


944650 


113 


731444 


504 


26^556 


41 


20 


676328 


390 


944582 


114 


731746 


504 


268254 


40 


SI 


9-676562 


390 


9-944514 


114 


9-732048 


504 


10-2679.52 


3f 


22 


676796 


390 


944446 


114 


732351 


503 


267649 


38 


23 


677030 


390 


944377 


114 


732653 


503 


267.347 


37 


24 


677264 


389 


944309 


114 


732955 


503 


267045 


36 


2.'i 


677498 


389 


944241 


114 


733257 


503 


266743 


35 


2f) 


677731 


389 


944172 


114 


733558 


503 


266442 


34 


27 


677964 


388 


944104 


114 


733860 


502 


266140 


33 


28 


678197 


388 


944036 


214 


734162 


502 


265838 


32 


29 


678430 


388 


^43967 


;i4 


734463 


502 


265537 


31 


30 


678663 


388 


943899 


.14 


734764 


502 


265236 


30 


31 


9-678895 


387 


P 943830 


114 


9-735066 


502 


10-264934 


29 


32 


679128 


387 


943761 


114 


735367 


502 


264633 


28 


33 


679360 


387 


943693 


115 


735668 


501 


2643.32 


27 


34 


679592 


387 


943624 


115 


735969 


501 


264031 


26 


35 


679824 


386 


043555 


115 


736269 


501 


263731 


25 


36 


680056 


386 


9434H6 


115 


736570 


501 


263430 


24 


37 


680288 


386 


94.3417 


115 


736871 


501 


263129 


23 


38 


6805 J 9 


385 


943.348 


115 


737171 


500 


262829 


22 


39 


680750 


385 


943279 


115 


737471 • 


500 


26'i529 


21 


40 


680982 


385 


943210 


115 


737771 


500 


262229 


20 


41 


9-681213 


385 


9-943141 


115 


9-738071 


500 


10-261929 


19 


42 


681443 


384 


943072 


115 


438371 


500 


261629 


18 


43 


681674 


384 


943003 


115 


738671 


499 


261.329 


17 


44 


681905 


384 


942934 


115 


738971 


499 


26)()'J9 


16 


45 


68213.5 


384 


942864 


115 


739271 


499 


260729 


15 


46 


682365 


383 


942795 


1J6 


739570 


499 


260430 


14 


47 


68ii595 


383 


942726 


116 


739870 


499 


2601.30 


13 


48 


6H2825 


383 


942656 


116 


740169 


499 


259831 


12 


49 


683055 


383 


942587 


116 


740468 


498 


259532 


11 


50 


683284 


382 


942517 


116 


740767 


498 


259233 


10 


51 


9-683514 


382 


9-942448 


116 


9-741066 


498 


10-2.58934 


9 


52 


683743 


382 


942378 


116 


741365 


498 


2.58635 


8 


53 


683972 


382 


942308 


116 


741664 


498 


258336 


7 


54 


684201 


381 


942239 


316 


741962 


497 


2.58038 


6 


55 


684430 


381 


942169 


116 


742261 


497 


257739 


5 


56 


684658 


381 


942099 


116 


742559 


497 


2.57441 


4 


57 


684887 


380 


942029 


116 


7428.58 


497 


2.57142 


3 


58 


685115 


380 


941959 


116 


7431.56 


497 


256H44 


2 


59 


685343 


380 


941889 


117 


743454 


497 


2.56.546 


1 


60 


685571 


380 


941819 


117 


743752 


496 


256248 






I Tftog. I M. 



LOGARITHMIC SIXES, CO SIXES, ETC. (20 Degrees.) 227 



M. I 

i! 



I Cosine | D. | Tang. 



I Cotang. 



9-68.5571 


380 


9-941819 


117 


9-7437.52 


496 


10-256248 


685799 


379 


941749 


117 


744050 


496 


255950 


68(j()27 


379 


941679 


117 


744348 


496 


255652 


686^254 


379 


941609 


117 


744645 


496 


255355 


68tt485> 


379 


941539 


117 


744943 


496 


255057 


68b7U9 


378 


94141)9 


117 


745240 


496 


254760 


686936 


378 


941398 


117 


745538 


495 


254462 


687J63 


378 


941328 


117 


745835 


495 


254165 


687389 


378 


941258 


117 


746132 


495 


253868 


687616 


377 


941187 


117 


746429 


495 


253571 


6M7843 


377 


941117 


117 


746726 


495 


253274 


9-688069 


377 


9-941046 


118 


9-747023 


494 


10-252977 


6W295 


377 


940975 


118 


747319 


494 


2.r2C81 


^Kyil 


376 


940905 


118 


747616 


494 


252384 


6887-17 


376 


940834 


118 


747913 


494 


252087 


68^^972 


3T6 


940763 


118 


748209 


494 


251791 


689198 


376 


910693 


118 


748505 


493 


251495 


689423 


375 


940622 


118 


748801 


493 


251199 


689648 


375 


940551 


118 


749097 


493 


250903 


689873 


375 


940480 


118 


749393 


493 


250607 


690098 


375 


94041)9 


1J8 


749689 


493 


250311 


9-690323 


374 


9-9-10338 


118 


9-749985 


493 


10-2500I5 


690548 


374 


94U267 


118 


750281 


492 


249719 


690772 


374 


940196 


118 


750576 


492 


249424 


690996 


374 


940125 


119 


750872 


492 


2491-28 


69J-2-20 


373 


940054 


119 


751167 


492 


248833 


691444 


373 


939982 


119 


751462 


492 


248538 


691668 


373 


939911 


119 


751757 


492 


248243 


691892 


373 


939840 


119 


752052 


491 


247948 


692115 


372 


939768 


119 


752347 


491 


247653 


692339 


372 


939697 


119 


752642 


491 


247358 


9-692562 


372 


9-939625 


119 


9-752937 


491 


10-247063 


692785 


371 


939554 


119 


753231 


491 


246769 


6930(J8 


371 


939482 


119 


753526 


491 


246474 


693231 


371 


939410 


119 


753820 


490 


246180 


693453 


371 


939339 


119 


754115 


490 


245885 


693676 


370 


939267 


120 


754409 


490 


245.591 


693898 


370 


939195 


120 


754703 


490 


24.5297 


694120 


370 


939123 


120 


754997 


490 


245003 


694342 


370 


939052 


120 


755291 


490 


244709 


694564 


369 


938980 


120 


755585 


489 


244415 


9-69.1786 


369 


9-938908 


i20 


9-755878 


489 


10-244122 


695007 


369 


938836 


J 20 


756172 


489 


243828 


69.5229 


369 


938763 


120 


756465 


489 


243535 


695450 


368 


938691 


190 


756759 


489 


243241 


695671 


368 


938619 


1-^-u 


757052 


489 


242948 


695892 


368 


938.547 


120 


757345 


488 


242655 


696113 


368 


938475 


120 


757638 


488 


242362 


696334 


367 


938402 


121 


757931 


488 


242069 


696554 


367 


938330 


121 


7.58224 


488 


241776 


696775 


367 


938258 


121 


758517 


488 


241483 


9-696995 


367 


9-938185 


121 


9-758810 


488 


10-241190 


697215 


366 


938113 


121 


759102 


487 


240898 


697435 


366 


938040 


121 


759395 


487 


240605 


697654 


366 


937967 


121 


759'>87 


487 


240313 


697874 


366 


937895 


121 


759979 


487 


240021 


698094 


365 


9378-22 


121 


760272 


4S7 


239728 


698313 


365 


937749 


121 


760564 


487 


239436 


698532 


365 


937078 


121 


760856 


486 


239144 


698751 


365 


93-004 


121 


761148 


486 


2388.52 


698970 


364 


937J31 


121 


761439 


486 


23856] 



Q??!RP I 



[ Sm 



I I Cotang. 

§9 Pegre§|, 



?»o?. I ¥^ 



228 (30 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. 


Sine 


D. 


Cosine 


D. 


1 Tang. 


D. 


Cotang. 







9-698970 


364 


9937531 


121 


9-761439 


486 


10-238561 


60 


1 


G99189 


364 


937458 


122 


761731 


486 


238269 


59 


2 


699407 


364 


937385 


122 


762023 


486 


237977 


58 


3 


699626 


364 


937312 


122 


762314 


486 


237686 


57 


4 


699844 


363 


937238 


122 


762606 


485 


237394 


56 


5 


700062 


363 


937165 


122 


762897 


485 


237103 


55 


6 


700280 


363 


937092 


122 


763188 


485 


236812 


54 


7 


700498 


363 


937019 


122 


763479 


485 


236521 


53 


8 


700716 


363 


936946 


122 


763770 


485 


236230 


52 


9 


700933 


362 


936872 


122 


764061 


485 


235939 


51 


30 


701151 


362 


936799 


122 


764352 


484 


235<J48 


50 


11 


9-701368 


362 


9-936725 


122 


9-764643 


484 


10-2353.57 


49 


12 


701585 


362 


936652 


123 


764933 


484 


235067 


48 


13 


701802 


361 


936578 


123 


765224 


484 


234776 


47 


14 


702019 


361 


936505 


123 


76.5514 


484 


234486 


46 


15 


702236 


361 


936431 


123 


765805 


484 


234195 


45 


16 


702452 


361 


936357 


123 


766095 


484 


233905 


44 


17 


702669 


360 


936284 


123 


766385 


483 


233615 


43 


18 


702885 


360 


936210 


1-^3 


766675 


483 


233325 


42 


19 


703101 


3H0 


936136 


123 


766965 


483 


233035 


41 


20 


703317 


360 


936062 


123 


767255 


483 


232745 


40 


21 


9-703533 


359 


9-935988 


123 


9-767545 


483 


10-2324.55 


39 


22 


703749 


359 


935914 


123 


767834 


483 


232166 


38 


23 


703964 


359 


935840 


123 


768124 


482 


231876 


37 


24 


7041. 9 


359 


935766 


124 


768413 


482 


231.587 


36 


25 


704395 


359 


935692 


124 


768703 


482 


231297 


35 


26 


704610 


358 


935618 


124 


7C8992 


482 


231008 


34 


27 


704825 


358 


935543 


124 


709281 


482 


230719 


33 


28 


705040 


358 


935469 


124 


769570 


482 


230430 


32 


29 


705254 


358 


935:?95 


124 


7G98(-0 


481 


230140 


31 


30 


705469 


357 


935:)2(J 


124 


770148 


481 


229852 


30 


31 


9-705683 


357 


9-935246 


124 


9-7704:57 


481 


10-229563 


29 


32 


705898 


3.17 


935171 


124 


77!;726 


481 


-229274 


■28 


33 


70(;|l'2 


357 


935097 


124 


7710J5 


481 


22F985 


27 


34 


7063'2(; 


356 


935022 


124 


771.303 


481 


228097 


26 


35 


706539 


356 


934948 


124 


771592 


481 


228408 


25 


36 


706753 


356 


934873 


124 


771880 


480 


228120 


24 


37 


706967 


356 


934798 


125 


772168 


480 


227832 


23 


38 


707180 


355 


934723 


125 


7724.57 


480 


227543 


22 


39 


707393 


355 


934649 


125 


772745 


480 


227255 


21 


40 


707606 


355 


934574 


125 


773033 


480 


22<-)967 


20 


41 


9-707819 


355 


9-934499 


125 


9-773321 


480 


10-226679 


19 


42 


708032 


354 


934424 


125 


773608 


479 


226392 


18 


43 


708245 


354 


93434!) 


125 


773896 


479 


226104 


17 


44 


708458 


354 


934274 


125 


774184 


479 


22.5816 


16 


45 


708670 


354 


934199 


125 


774471 


479 


225.529 


15 


40 


708882 


353 


934J23 


125 


7747.59 


479 


225241 


14 


47 


709094 


353 


934043 


125 


775046 


479 


224954 


13 


48 


709306 


353 


933973 


125 


775333 


479 


224667 


12 


49 


709518 


353 


933898 


"126 


775621 


478 


224379 


11 


50 


709730 


353 


933822 


126 


775908 


478 


224092 


10 


51 


9-709941 


352 


9-933747 


126 


9-776195 


478 


10-223805 


9 


52 


710153 


352 


93:^67 1 


126 


776482 


478 


223518 


8 


53 


710364 


352 


933596 


126 


77t)769 


478 


223231 


7 


54 


710575 


352 


!t:(;f5-J() 


126 


777055 


478 


222945 


6 


55 


710786 


351 


93:W45 


126 


777342 


478 


222658 


5 


56 


710997 


351 


933369 


126 


777628 


477 


222372 


4 


57 


711208 


351 


933293 


126 


777915 


477 


222085 


3 


58 


711419 


351 


933217 


126 


778201 


477 


221799 


2 


59 


711629 


350 


933141 


126 


778487 


477 


2215 12 


I 


GO 


711839 


350 


933066 


126 


778774 


477 


221226 






1 Coiine | 



I CotanR. I 



Tang. J M 



fi9 Degrees. 



LOGAIilTHMIC SfXES, CO-^lX/i-'^. ETC. (31 Degrees.) 22fi 



M. 


Sine 


D. 1 


Cosine 


D. 1 


Trng. 1 


D. 1 


Colang. I 







9^11839 


350 


9-933066 


126 


9-778774 


477 


10-221226 


60 


1 


712050 


350 


932990 


127 


779060 


477 


220940 


59 


2 


712260 


350 


932914 


127 


779346 


476 


220654 


58 


3 


712469 


349 


932838 


]27 


779632 


476 


220368 


57 


4 


712679 


349 


932762 


127 


779918 


476 


220082 


56 


5 


712889 


349 


932685 


127 


780203 


476 


219797 


55 


6 


713098 


349 


932609 


127 


780489 


476 


219511 


54 


7 


713303 


349 


932533 


127 


780775 


476 


219225 


53 


8 


713517 


348 


932457 


127 


781060 


476 


218940 


52 


9 


713726 


348 


932380 


127 


781 :M6 


475 


218654 


51 


10 


713935 


348 


932304 


127 


781631 


475 


2J8369 


50 


11 


9-714144 


348 


9-932228 


127 


9-781916 


475 


10-218084 


49 


12 


714352 


347 


932151 


127 


782201 


475 


2r7799 


48 


i:} 


714561 


347 


932075 


128 


782486 


475 


217514 


47 


14 


714769 


347 


931998 


128 


782771 


475 


217229 


40 


15 


714978 


347 


931921 


12.8 


783056 


475 


216944 


45 


]f» 


715186 


347 


931845 


123 


783341 


475 


216059 


44 


17 


715394 


346 


931768 


123 


783626 


474 


216374 


43 


18 


715602 


346 


93 1691 


128 


783910 


474 


216090 


42 


19 


715809 


346 


931014 


128 


784195 


474 


215805 


41 


20 


710017 


346 


931537 


128 


784479 


474 


21552J 


40 


21 


9-716224 


345 


9-931460 


128 


9-784764 


474 


10-215236 


39 


22 


716432 


345 


931383 


128 


785048 


474 


214952 


38 


23 


710639 


345 


931306 


128 


7P5332 


473 


214C68 


37 


24 


71G346 


345 


93122D 


129 


785616 


473 


214384 


36 


25 


717053 


345 


931152 


129 


785900 


473 


214100 


35 


26 


717259 


344 


931075 


129 


786184 


473 


213816 


34 


27 


717466 


344 


930998 


129 


786468 


473 


213532 


33 


28 


717673 


344 


930921 


129 


786752 


473 


213248 


32 


29 


717879 


344 


930843 


129 


78r036 


473 


212964 


31 


30 


718085 


343 


930766 


129 


787319 


472 


212681 


30 


31 


9-7J8291 


343 


9-930688 


129 


9-787603 


472 


10-21239/ 


29 


32 


718497 


343 


930611 


129 


787886 


472 


212114 


28 


33 


718703 


343 


930533 


129 


788170 


472 


211830 


27 


34 


7189C9 


343 


930456 


129 


788453 


472 


211547 


20 


35 


719114 


342 


930378 


129 


788736 


472 


211204 


25 


36 


719320 


342 


930300 


130 


789019 


472 


210981 


24 


37 


719525 


342 


930223 


130 


789302 


471 


210C38 


23 


38 


719730 


342 


930145 


130 


789585 


471 


210415 


22 


J9 


719935 


341 


930067 


130 


789868 


471 


210132 


21 


40 


720140 


341 


929989 


130 


790151 


471 


209849 


20 


41 


9-720345 


341 


9-929911 


130 


9-790433 


471 


10-209567 


19 


42 


720549 


341 


929833 


130 


790716 


471 


209284 


18 


43 


720754 


340 


929755 


130 


790933 


471 


209001 


17 


44 


720958 


340 


929677 


130 


791231 


471 


208719 


16 


45 


721162 


340 


929599 


130 


79i.-;G3 


470 


208437 


15 


46 


721366 


340 


929521 


130 


7i::846 


470 


208154 


14 


47 


721570 


340 


929442 


130 


7:!2I28 


470 


207872 


13 


48 


721774 


339 


929364 


131 


792410 


470 


207590 


J2 


49 


721978 


339 


929286 


131 


792692 


470 


207308 


11 


50 


722181 


339 


929207 


131 


792974 


470 


207026 


10 


51 


!i-722385 


339 


9-929129 


131 


9-793256 


470 


10-206744 


9 


52 


722588 


339 


929050 


131 


793538 


469 


206462 


8 


53 


722791 


338 


928972 


131 


793819 


469 


206181 


7 


54 


722994 


338 


928893 


131 


794101 


469 


205899 


6 


55 


72:1197 


338 


928815 


131 


794383 


469 


205317 


5 


56 


723400 


338 


928736 


131 


794664 


469 


203336 


4 


57 


723603 


337 


928657 


131 


794945 


469 


205055 


3 


58 


723805 


337 


928578 


131 


795227 


469 


204773 


2 


59 


724007 


337 


928499 


131 


795508 


468 


204492 


1 


SO 


t24210 


337 


928420 


131 


795789 


468 


204211 






I Cosine \ 



I Sine 



I Coianf. I 



I Tang, ] AL 



58 Decrees. 



230 (32 Degrees.) LOGARJTITMIC SINES, COSINES, ETC. 



M. 1 


Sine 


D. 


Cosine 


D. 


Tail?. 


D. 


Cotans-. 







9-724210 


337 


9-928420 


132 


9-795789 


468 


10-20421] 


6(1 


1 


724412 


337 


928342 


132 


796070 


468 


203i.30 


59 


2 


724614 


3:i6 


92H263 


132 


796351 


468 


203649 


5!^ 


3 


724816 


336 


928183 


132 


79<>632 


468 


203368 


57 


4 


725017 


336 


928104 


132 


796til3 


468 


203087 


56 


5 


725219 


336 


928025 


132 


797194 


468 


202806 


55 


6 


725420 


335 


927946 


132 


797475 


468 


202525 


54 


7 


725622 


335 


927867 


132 


797755 


468 


202245 


53 


8 


725823 


335 


927787 


132 


798036 


467 


201964 


52 


9 


726024 


335 


927708 


132 


798316 


467 


201684 


51 


10 


726225 


335 


927629 


132 


798596 


467 


201404 


5(f 


11 


9-726426 


334 


9-927549 


132 


9-798877 


467 


10-201123 


49 


12 


726026 


334 


927470 


133 


799157 


467 


200843 


48 


13 


726827 


334 


927390 


133 


799437 


467 


200563 


47 


14 


727027 


334 


927310 


133 


799717 


467 


200283 


46 


15 


727228 


334 


927231 


133 


799997 


466 


200003 


45 


16 


727428 


333 


927151 


133 


800277 


466 


199723 


44 


17 


727628 


333 


927071 


133 


800557 


466 


199443 


43 


18 


727828 


333 


926991 


133 


800836 


466 


199164 


42 


19 


728C27 


333 


926911 


133 


801116 


466 


198884 


41 


20 


728227 


333 


926831 


133 


801396 


4()6 


198604 


40 


21 


9-728427 


332 


9-926751 


133 


9-801675 


466 


10-198325 


39 


23 


728626 


332 


926671 


133 


801955 


466 


198045 


38 


23 


728825 


332 


926591 


133 


802234 


465 


197766 


37 


24 


729024 


332 


920511 


134 


802513 


465 


197487 


36 


35 


729223 


331 


926431 


134 


802792 


465 


197208 


35 


26 


729422 


331 


926351 


134 


803072 


465 


196928 


34 


27 


729621 


331 


926270 


134 


803351 


465 


196649 


33 


28 


729820 


331 


926190 


134 


803630 


465 


196370 


32 


29 


730018 


330 


926110 


134 


803908 


465 


196092 


31 


30 


730216 


330 


926029 


134 


804187 


465 


195813 


30 


31 


9-730415 


330 


9-925949 


134 


9-804466 


464 


10-195534 


29 


32 


730613 


330 


925868 


134 


804745 


464 


195255 


28 


33 


730811 


330 


925788 


134 


805023 


464 


194977 


27 


34 


731009 


329 


925707 


134 


. 805302 


464 


194698 


96 


35 


731200 


329 


925626 


134 


805580 


464 


194420 


25 


3G 


731404 


329 


925545 


135 


805859 


464 


194141 


24 


37 


731602 


329 


925465 


135 


806 137 


464 


193863 


23 


38 


7317'jy 


329 


925384 


135 


8C6415 


463 


193585 


22 


29 


731096 


328 


925303 


135 


806693 


463 


193307 


21 


40 


732193 


328 


925222 


135 


806971 


463 


193029 


20 


41 


9-732300 


328 


9-925141 


135 


9-807249 


463 


10-192751 


19 


42 


732587 


328 


925060 


135 


807527 


463 


192473 


18 


43 


7:^2784 


328 


924979 


135 


807805 


4G3 


192105 


17 


44 


732980 


327 


924897 


135 


808083 


463 


191C17 


18 


45 


733177 


327 


924816 


135 


808361 


463 


191639 


15 


4(5 


733373 


327 


924735 


136 


808638 


4G2 


1913C2 


14 


47 


733539 


327 


924G54 


136 


808916 


462 


191C84 


13 


48 


7337G5 


327 


924572 


136 - 


809193 


462 


190807 


12 


49 


733961 


326 


924491 


136 


809471 


462 


190529 


11 


50 


734157 


326 


924409 


136 


809748 


462 


190252 


iO 


51 


9-734353 


326 


9-924328 


136 


9-810025 


462 


10-189975 


9 


52 


73^1549 


32G 


92-^2^6 


136 


810302 


4G2 


183698 


b 


53 


734744 


325 


924164 


136 


813580 


462 


189429 


7 


54 


734939 


325 


924083 


136 


810857 


462 


180143 


6 


55 


735135 


325 


924001 


136 


811134 


461 


188866 


5 


56 


735330 


325 


923919 


136 


811410 


461 


188590 


4 


57 


735525 


325 


923837 


136 


811687 


461 


188313 


2 


58 


735719 


324 


923755 


137 


811964 


461 


188036 


2 


59 


735914 


324 


923673 


137 


812241 


461 


187759 


1 


fiO 


736109 


324 


923591 


137 


812517 


461 


187483 







) Cosine 


1 


1 Sine 

i 


1 
>7Degre 


Cotang. 

es. 


1 


fang. 


M. 



^OGAA'/TI/J/IC SnVFS, COSINES, ETC. (33 Degrees.) 231 



Sine 


D. 


Cosine 


D. 


Tang. 


1 D. 


Cotanff. 1 




!i-736109 


324 


9-923591 


137 


9-812517 


461 


10-187482 


00 


7:56303 


324 


923509 


137 


812794 


461 


1872G6 


59 


7304 98 


324 


923427 


137 


813070 


461 


1869.30 


^^ 


736692 


323 


923345 


137 


813347 


460 


186653 


57 


736886 


323 


923263 


137 


813623 


460 


186377 


56 


737080 


323 


923181 


137 


813899 


460 


180101 


55 


737274 


323 


923098 


137 


814175 


460 


185825 


54 


737467 


323 


923016 


137 


814452 


460 


185548 


53 


737661 


322 


922933 


137 


814728 


460 


185272 


52 


737855 


322 


922851 


137 


815004 


460 


184996 


51 


738048 


322 


922768 


138 


815279 


460 


184721 


5C 


9-738241 


322 


9-922686 


138 


9-815555 


459 


10 184445 


49 


738434 


322 


922(i03 


138 


815831 


459 


1841()9 


48 


738627 


321 


922520 


138 


816107 


459 


183893 


47 


738820 


321 


922438 


138 


816382 


459 


183618 


46 


739013 


321 


922355 


138 


810658 


459 


183342 


45 


739206 


321 


922272 


138 


816933 


459 


183067 


44 


739398 


321 


922189 


138 


817209 


459 


182791 


43 


739590 


320 


922106 


138 


817484 


459 


182516 


42 


739783 


320 


922023 


138 


817759 


459 


182241 


41 


739975 


320 


921940 


138 


818035 


458 


181965 


40 


9-740167 


320 


9-921857 


139 


9-818310 


458 


10-181690 


39 


740359 


320 


921774 


139 


818585 


458 


181415 


38 


740550 


319 


921691 


139 


818860 


458 


181140 


37 


740742 


319 


921607 


139 


819135 


458 


180865 


30 


740934 


319 


921524 


139 


819410 


458 


180590 


35 


741J25 


319 


921441 


139 


819684 


458 


180316 


34 


741316 


319 


921357 


139 


819959 


458 


180041 


33 


7415C8 


318 


921274 


139 


820234 


458 


179766 


:^ 


741699 


318 


921 190 


139 


820508 


457 


J 79492 


3; 


74J889 


318 


921107 


139 


820783 


457 


179217 


3U 


9-742080 


318 


9-921023 


139 


9-821057 


457 


10-178943 


29 


742271 


318 


920939 


140 


821332 


457 


178668 


28 


742462 


317 


92C856 


140 


821606 


457 


178394 


27 


742652 


317 


920772 


140 


821880 


457 


178120 


26 


742842 


317 


92C088 


140 


822154 


457 


177846 


25 


743033 


317 


92CGC4 


140 


822429 


457 


177571 


24 


743223 


317 


92C520 


140 


822703 


457 


177297 


23 


743413 


310 


92C436 


140 


822977 


456 


177023 


22 


743602 


316 


920352 


140 


823250 


456 


176750 


21 


743792 


316 


920268 


140 


823524 


456 


170476 


20 


9-743982 


316 


9-920184 


140 


9-823798 


456 


10-176202 


19 


744171 


316 


920099 


140 


824072 


456 


175928 


18 


744361 


315 


92U015 


140 


824345 


456 


175655 


17 


744550 


315 


919931 


141 


824019 


456 


17i381 


16 


744739 


315 


919846 


141 


824893 


456 


175107 


15 


744928 


315 


919762 


141 


825160 


456 


174834 


14 


745117 


315 


919677 


141 


825439 


455 


174561 


13 


745306 


314 


919593 


141 


825713 


455 


174287 


12 


745494 


314 


919508 


141 


825986 


455 


174014 


11 


745683 


314 


919434 


141 


826259 


455 


173741 


IC 


9-745871 


314 


.9-919339 


141 


9-826532 


455 


10-173468 


9 


746059 


314 


919254 


141 


826805 


455 


173195 


8 


746248 


313 


919169 


141 


827078 


455 


172922 


7 


746436 


313 


919085 


141 


827351 


455 


172649 


6 


746624 


313 


919000 


141 


827624 


455 


172376 


5 


746812 


313 


918915 


142 


82'/89V 


454 


172103 


4 


746999 


313 


918830 


142 


828170 


454 


171830 


3 


747187 


312 


918745 


142 


828442 


454 


171558 


2 


747374 


312 


918659 


142 


828715 


454 


171285 


1 


747562 


312 


918574 


142 


828987 


454 


171013 


* 



1 CcMine ( 



I J Cotiuag. I 



I 1»*>B- lift 



232 (34 Degrees.) LOGARITHMIC RINES, COSIMIJS, ETC. 



M. 


1 Sine 


i D- 


1 Cosine 


I D. 


Tar-. 


1 D. 


Colang. 







9-747562 


312 


9-918574 


142 


9-828987 


454 


10-171013 


~60 


1 


7-1774J 


3J2 


918489 


142 


829260 


454 


170740 


59 


2 


747936 


312 


918404 


142 


829532 


454 


170468 


58 


3 


748123 


311 


918318 


142 


829805 


454 


170195 


57 


4 


7483 10 


311 


918233 


142 


83tj077 


454 


169923 


56 


5 


748497 


311 


918147 


142 


830349 


453 


169651 


55 


6 


748^83 


311 


i* 18062 


J 42 


83(J621 


453 


169379 


54 


7 


748870 


311 


917976 


143 


830893 


453 


169107 


53 


8 


74905G 


310 


917891 


143 


831165 


453 


168835 


52 


9 


749243 


310 


917805 


143 


831437 


453 


168563 


51 


10 


749423 


310 


917719 


143 


831709 


453 


J68291 


50 


11 


9-749615 


310 


9-917034 


143 


9-831981 


453 


10-168019 


49 


J2 


749801 


310 


917548 


143 


832253 


453 


167747 


48 


13 


749987 


309 


917462 


143 


832525 


453 


167475 


47 


14 


750172 


309 


917376 


143 


832796 


453 


167204 


46 


15 


750358 


309 


9 J 7290 


143 


833068 


452 


J 66932 


45 


.10 


750543 


309 


917204 


143 


833339 


452 


166661 


44 


17 


750729 


309 


917118 


144 


833611 


452 


166389 


43 


18 


750914 


308 


917032 


144 


833882 


452 


166118 


42 


19 


751099 


308 


916946 


144 


834154 


452 


165846 


41 


20 


751284 


308 


916859 


144 


834425 


452 


165575 


40 


21 


9-751469 


308 


9-916773 


144 


9-834696 


452 


10-165304 


39 


22 


751G54 


308 


916687 


144 


834967 


452 


165033 


38 


23 


751839 


308 


91G600 


144 


835238 


452 


l{i4762 


37 


24 


752U23 


307 


910514 


144 


835509 


452 


1C4491 


36 


25 


752208 


307 


916427 


144 


835780 


451 


1M2-20 


35 


26 


752392 


307 


916341 


144 


830051 


451 


163949 


34 


27 


752576 


307 


91o254 


144 


836322 


451 


103678 


33 


28 


752760 


307 


916167 


145 


836593 


451 


163407 


32 


29 


752944 


306 


916081 


145 


836864 


451 


163136 


31 


30 


753128 


306 


915994 


145 


837134 


451 


162866 


30 


31 


9-753312 


306 


9-915907 


145 


9-837405 


451 


10-162595 


29 


32 


7u3495 


306 


915820 


145 


837075 


451 


162325 


28 


33 


753679 


306 


915733 


145 


837946 


451 


162054 


27 


34 


753862 


305 


915646 


145 


838216 


451 


1G1784 


26 


35 


754046 


305 


915559 


145 


838487 


450 


161513 


25 


36 


754229 


305 


915472 


145 


838757 


450 


161243 


24 


37 


754412 


305 


915385 


145 


839027 


450 


1G0973 


23 


38 


754595 


3U5 


915297 


J45 


839297 


450 


160703 


22 


39 


754778 


304 


915210 


145 


839568 


450 


160432 


21 


40 


754960 


304 


915123 


146 


839838 


450 


160162 


20 


41 


9-755143 


304 


9-915035 


146 


9-840108 


450 


10159892 


19 


42 


755326 


304 


914948 


146 


840378 


450 


159622 


18 


43 


755508 


304 


914860 


146 


840647 


450 


159353 


17 


44 


755690 


304 


914773 


146 


840917 


449 


159083 


16 


45 


755872 


303 


914685 


146 


841187 


449 


158813 


15 


46 


756054 


303 


914598 


146 


841457 


449 


158543 


14 


47 


756236 


303 


914510 


146 


841726 


449 


158274 


13 


48 


756418 


303 


914422 


146 


841996 


449 


158004 


12 


49 


756600 


303 


914334 


146 


842266 


449 


J 57734 


11 


50 


756782 


302 


914246 


147 


842535 


449 


157465 


10 


51 


9-756963 


302 


9-914158 


147 


9-842805 


449 


10157195 


9 


52 


757144 


302 


914070 


147 


843074 


449 


156926 


8 


53 


757326 


302 


9 J 3982 


147 


843343 


449 


156657 


7 


54 


757507 


302 


913894 


147 


843612 


449 


156388 


6 


55 


757688 


301 


9 J 3806 


147 


843882 


448 


156 J 18 


5 


56 


757869 


301 


913718 


147 


844151 


448 


155849 


4 


57 


758050 


301 


913630 


147 


844420 


448 


155580 


3 


58 


758230 


301 


913541 


147 


844689 


448 


155311 


2 


5tt 


758411 


301 


913453 


147 


844958 


448 


-155042 


1 


60 


758591 


301 


913365 


147 


845227 


448 


154773 






Cosine | 



\ Cotong. I 



I Tang. I 



55 Degrees. 



LOGARITHMIC SIXES, COSIXES, ETC. (35 Degrees.) 23S 



I Cosine 



I ThM^T 



I Colanjf. 






9-758591 


301 


9-913365 


147 


9.845227 


448 


10-154773 


60 


1 


758772 


300 


913276 


147 


W4.")496 


448 


154.504 


59 




75895-2 


300 


913187 


148 


b45764 


448 


1 54:^30 


58 


3 


753132 


300 


913099 


148 


S46033 


448 


153967 


57 


4 


75D312 


300 


913010 


14ti 


6-,u.jo2 


448 


153ujri 


50 


5 


759452 


300 


912922 


148 


84()570 


447 


153430 


55 


6 


759G72 


2yy 


912833 


148 


846639 


447 


153161 


54 


7 


759852 


2j9 


912744 


148 


847107 


447 


152893 


53 


8 


76C031 


299 


912G55 


148 


847.376 


447 


152G24 


52 


9 


7C0211 


299 


9125GG 


148 


S47G-;4 


447 


15235G 


51 


10 


7G03D0 


299 


912477 


143 


847913 


447 


152387 


50 


II 


9-7G05G9 


298 


9-912388 


148 


9.8^8181 


447 


10151819 


49 


12 


7G0748 


2;,8 


912-299 


149 


Si'6-WJ 


447 


151551 


4H 


13 


7GG'J27 


298 


912210 


149 


C-:87JI7 


447 


15I2S3 


47 


It 


7C110G 


298 


912121 


149 


84Si}oG 


447 


151014 


40 


1.-. 


7Gi-235 


298 


912031 


149 


84D254 


447 


15074G 


45 


16 


7Gin;i 


■2J3 


91 1042 


149 


84L522 


447 


150478 


44 


17 


7G1G42 


297 


911853 


149 


8^0790 


446 


150210 


43 


18 


7G182I 


297 


9117G3 


149 


850358 


446 


149942 


42 


lii 


7GlJ9il 


297 


91 1074 


149 


S5G325 


446 


149075 


41 


•20 


7G2177 


297 


911584 


149 


85G593 


446 


149407 


40 


21 


97G235G 


297 


9-911495 


149 


9850:^61 


446 


10-149139 


30 


22 


7G2534 


296 


911405 


149 


851129 


446 


148871 


38 


23 


7G2712 


29G 


911315 


150 


851396 


44G 


148G04 


37 


24 


7G2889 


29G 


9J122G 


150 


8.:jCG4 


44G 


14S:^3G 


30 


2.) 


7(i30G7 


296 


9Jlio6 


150 


851^31 


446 


14S.G9 


35 


•2tj 


7t)3245 


2.'iG 


9iliMG 


150 


852199 


446 


147801 


34 


■J 7 


703422 


2;>ti 


9JU956 


150 


852466 


446 


147534 


33 


.'■s 


7l)3filiO 


2j.') 


9W866 


15U 


8.52733 


445 


147267 


32 


29 


7(i3777 


2J5 


910776 


150 


85;i001 


445 


146999 


31 


30 


7.13954 


295 


916G86 


15 J 


853268 


445 


146732 


30 


31 


9-7G4131 


295 


9-910596 


150 


9-853535 


445 


10-146465 


29 


3-2 


7G4308 


295 


9i;}5GG 


15 J 


8..3t)J2 


445 


146198 


28 


33 


7G4-I85 


294 


910415 


150 


8540v39 


445 


145931 


27 


34 


7G4(iG2 


294 


910325 


151 


8543.J6 


445 


145664 


26 


35 


7<i4838 


294 


910235 


151 


S54()03 


445 


1453'j7 


25 


36 


7(>5()15 


294 


910144 


151 


854870 


445 


145130 


24 


37 


765191 


294 


910054 


151 


855137 


445 


144863 


23 


38 


7G5367 


294 


909963 


151 


8554U4 


445 


144.596 


22 


39 


705544 


293 


909873 


151 


8.55671 


444 


1443-29 


21 


4G 


765720 


293 


909782 


151 


8559.i8 


444 


144062 


20 


41 


9-7G5896 


293 


9-909691 


151 


9-85G2V.4 


444 


10-14379G 


19 


42 


766072 


293 


909601 


151 


850471 


444 


1435-29 


18 


43 


766247 


293 


909510 


151 


85G737 


444 


1432G3 


17 


44 


766423 


293 


909419 


151 


8570;)4 


444 


142996 


6 


45 


766598 


2i)2 


909328 


152 


S57270 


444 


142730 


5 


46 


766774 


2:12 


909237 


152 


857537 


444 


1424 G3 


4 


47 


76(i949 


2l)2 


909 J 46 


152 


8578C3 


444 


14-2 J i)7 


13 


48 


767124 


2i)2 


909055 


152 


858069 


444 


14193! 


12 


49 


767300 


292 


908964 


152 


858336 


444 


141664 


U 


50 


767475 


291 


90P873 


152 


858G02 


443 


141398 


10 


51 


9-7G7649 


291 


9-908781 


152 


9-858868 


443 


10-141132 


9 


52 


767824 


291 


90P690 


152 


8.59134 


443 


140866 


8 


53 


767999 


291 


908599 


152 


859400 


443 


140600 


7 


54 


768173 


291 


903507 


152 


859666 


443 


140334 


6 


55 


768348 


290 


908416 


153 


859932 


443 


140068 


5 


56 


768522 


290 


908324 


153 


860198 


443 


139802 


4 


57 


768697 


290 


908233 


153 


860464 


443 


139536 


3 


58 


768871 


290 


908141 


153 


860730 


443 


139270 


2 


59 


769045 


290 


908049 


153 


860995 


443 


139005 


1 


60 


769219 


290 


907958 


153 


861261 


443 


138739 






I C(Mine I 



I Sine I 



Cotang. I 



I Tang. I M. 



54 Degrees. 



234 (36 Degr( 



LVC Mi I Til MIC SIX/.S, CO, SIXES, ETC 



M. 


Sine 


1 D. 





9-769219 


290 


1 


769393 


289 


2 


769566 


289 


3 


769740 


289 


4 


7699J3 


289 


5 


770087 


289 


G 


770260 


288 


7 


770433 


'288 


8 


770606 


2BS 





770779 


288 


i;) 


770952 


288 


11 


9 77-125 


288 


i-i 


771298 


287 


13 


771470 


287 


J4 


771643 


287 


]"> 


771815 


287 


Ifi 


771987 


287 


]7 


772159 


287 


16 


772331 


•286 


19 


772503 


2,^6 


20 


772675 


'J8i) 


21 


9-772847 


236 


22 


773018 


286 


23 


773190 


28G 


24 


773361 


285 


25 


773533 


285 


m 


773704 


285 


27 


773875 


285 


28 


774046 


285 


2;J 


774217 


285 


30 


774388 


284 


31 


9-774558 


284 


32 


774729 


284 


33 


774899 


284 


34 


775070 


284 


35 


775240 


284 


36 


775410 


283 


37 


775580 


283 


38 


775750 


283 


39 


775920 


283 


40 


776090 


283 


41 


9-77625C 


283 


42 


776429 


282 


43 


776598 


282 


44 


776768 


282 


45 


77(i937 


282 


46 


777106 


282 


47 


777275 


281 


48 


777444 


281 


49 


777613 


281 


50 


777781 


281 


51 


9-777950 


281 


52 


778J 19 


281 


53 


778287 


280 


54 


778455 


280 


55 


778624 


280 


56 


778792 


280 


57 


7789«0 


280 


58 


779128 


280 


59 


779295 


279 


60 


779463 


279 



Cosine 


D. 1 


Tang. 


D. 


Cotang. 


9~9~U7958 


153 


9-861261 


443 


10-138739 


907866 


153 


861527 


443 


138473 


907774 


153 


861792 


442 


138208 


907682 


253 


862058 


442 


137942 


907590 


J 53 


862323 


44j» 


137677 


9o7498 


153 


8625«9 


442 


137411 


907406 


153 


8t)2854 


442 


137146 


907314 


154 


86;;] 19 


442 


136881 


907222 


154 


863385 


442 


136615 


907129 


154 


863650 


442 


136350 


907037 


154 


863915 


442 


136(»85 


9-906945 


154 


9-864180 


442 


10-135820 


906852 


154 


864445 


442 


135555 


906760 


154 


864710 


442 


135290 


906667 


154 


864975 


441 


135025 


906575 


154 


865240 


441 


134700 


906482 


154 


865505 


441 


134495 


906389 


155 


865770 


441 


134230 


906296 


155 


866035 


441 


133965 


906204 


155 


866300 


441 


133700 


906111 


155 


866564 


441 


133436 


9-906018 


155 


9-866829 


441 


10-13317; 


905925 


155 


867094 


441 


132906 


905832 


155 


867358 


441 


'32642 


905739 


155 


867623 


441 


132377 


905645 


155 


8D7S87 


441 


132113 


905552 


155 


86815:^ 


440 


131848 


905459 


155 


868416 


440 


131584 


905366 


156 


86»d60 


440 


131320 


905272 


156 


868945 


440 


131055 


905179 


156 


86D209 


440 


130791 


9-905085 


156 


9-869473 


440 


10-130527 


904992 


156 


869737 


440 


]3026:{ 


904898 


156 


870001 


440 


J 29999 


904804 


156 


870265 


440 


129735 


904711 


156 


870o29 


440 


129471 


904617 


156 


87G793 


440 


129207 


904523 


156 


871057 


440 


128943 


904429 


157 


871321 


440 


128679 


904335 


357 


871585 


440 


128415 


90424] 


157 


871849 


439 


128151 


0-904147 


157 


9-872112 


439 


10-127888 


904053 


157 


872376 


439 


127624 


903959 


157 


872640 


439 


127360 


903864 


157 


872903 


439 


127097 


903770 


157 


873167 


439 


126833 


903676 


157 


873430 


439 


126570 


903581 


157 


873694 


439 


12»)306 


903487 


157 


873957 


439 


126043 


903392 


158 


874220 


439 


125780 


903298 


158 


874484 


439 


125516 


9-903203 


158 


9-874747 


439 


10-125253 


903108 


158 


875010 


439 


124990 


90:'(I14 


158 


875273 


438 


124727 


9i;29l9 


158 


875536 


438 


124464 


902824 


158 


875800 


438 


124200 


902729 


158 


876063 


438 


123937 


902634 


158 


876326 


438 


123674 


902539 


159 


876589 


438 


123411 


9112444 


159 


876851 


438 


12:n49 


902349 


159 


877114 


438 


122886 



I Cosine I 



Sine I I Cotang. 

©3 Pegrwg, 



Tws- 



LOGARITHMIC SINES, COSINES, ETC. (37 Degrees.) 235 



M. I 



Sine 

flf79463 

779631 

779798 
779966 
780133 
7H03U0 
780467 
78U634 
7808U1 
780968 
781J34 

9-781301 
781468 
781634 
781800 
781966 
781; 132 
782298 
782464 
782(;30 
782'/ 96 

9-782961 
783127 
783292 
783458 
783623 
783788 
783953 
784118 
784282 
784447 

9-784612 
784776 
784941 
785105 
785269 
785433 
785597 
785761 
785925 
786089 

9-786252 
786416 
786579 
786742 
786906 
7^57069 
7H7232 
787395 
787557 
787720 

9-787883 
788i;45 
7H8208 
7HH:rO 
7«H532 
78>'«i94 
788856 
789018 
789180 
789342 



279 
279 

279 
279 
279 
278 
278 
278 
278 
278 
278 
277 
277 
277 
277 
277 
277 
276 
276 
276 
276 
276 
276 
275 
275 
275 
275 



273 
273 
273 
273 
273 
272 
272 
272 
272 
272 
272 
271 
271 
271 
271 
271 
271 
271 

2-; I) 

270 
270 
270 
270 
270 



Cosine 

9-902349 
902253 
902158 
90-io63 
9(>1967 
90 J 872 
901776 
901681 
90 J 585 
901490 
901394 

9-901298 
901202 
901106 
901010 
900914 
900818 
900722 
900626 
900529 
900433 

9-900337 
900240 
900144 
900047 
899951 
899854 
899757 
899660 
8995(>4 
899467 

9-899370 
899273 
899176 
899078 
898981 
898884 
8D8787 
898680 
818592 
898494 

9-898397 
898299 
898202 
898104 
898006 
897908 
897810 
897712 
897614 
897516 

9-897418 
897320 
8H722-2 
)^:t7l-23 
897(K25 
89H926 
896828 
896729 
89663 1 
896532 



1 D. 


Tanj. 


D. 


Cotang. 




159 


9-877114 


438 


10-122886 


60 


159 


877377 


438 


122623 


59 


159 


877640 


438 


122360 


58 


159 


877903 


438 


122097 


57 


159 


878 J 65 


438 


121835 


56 


159 


878428 


438 


121572 


55 


159 


878691 


438 


121309 


.54 


159 


878953 


437 


121047 


53 


159 


879216 


437 


120784 


52 


159 


879478 


437 


120522 


51 


160 


879741 


437 


120259 


50 


160 


9-880003 


437 


10-119997 


49 


160 


880265 


437 


1 J 9735 


48 


160 


880528 


437 


119472 


47 


160 


880790 


437 


119210 


46 


160 


881052 


437 


118948 


45 


160 


88J314 


437 


118686 


44 


160 


881573 


437 


118424 


43 


160 


881839 


437 


118161 


42 


160 


882101 


4:n 


117899 


41 


161 


882363 


436 


117637 


40 


161 


9-882625 


436 


10117375 


39 


161 


882887 


436 


117113 


38 


iril 


883148 


436 


116852 


37 


1151 


883410 


436 


116590 


36 


161 


883672 


436 


116328 


35 


161 


883934 


436 


116066 


34 


161 


884 J 96 


436 


115804 


33 


161 


884457 


436 


115543 


32 


161 


884719 


436 


115281 


31 


162 


884980 


436 


115020 


30 


162 


9-885242 


436 


10-114758 


29 


162 


885503 


436 


114497 


28 


162 


885765 


436 


1 J 4235 


27 


162 


886026 


436 


113974 


26 


162 


886288 


436 


1137i2 


25 


162 


886549 


435 


113451 


24 


162 


886810 


435 


113190 


23 


102 


887072 


435 


112928 


22 


162 


887333 


435 


112667 


21 


163 


887594 


435 


112406 


20 


163 


9-P87855 


435 


10-112145 


19 


163 


888116 


435 


111884 


18 


163 


888377 


435 


111623 


17 


163 


888639 


435 


111361 


16 


163 


888900 


435 


111100 


15 


163 


889160 


435 


110840 


14 


163 


889422 


435 


110579 


13 


1()3 


889(;82 


435 


110318 


12 


163 


889943 


435 


110057 


11 


163 


890204 


434 


109796 


10 


164 


9-890465 


434 


10-109535 


9 


164 


890725 


434 


109275 


8 


164 


890986 


434 


109014 


7 


164 


891247 


434 


108753 


6 


164 


891507 


434 


108493 


5 


164 


891768 


434 


108232 


4 


164 


892028 


434 


107972 


3 


164 


892289 


434 


107711 


2 


164 


892549 


434 


107451 


1 


164 


892810 


434 


107190 






I ^vm,% I 



I Sine 



I Cotang, I 



I fSPf. 



P? Pegr??8, 



236 (38 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. 


Sine 


-D. 1 


Cosine 


D. 


Tang. 


D. 


Cotang. 







9-789342 


269 


9-896532 


164 


9-892810 


434 


10-107190 


60 


] 


789504 


269 


896433 


165 


893070 


434 


106930 


59 


2 


789665 


269 


896335 


165 


893331 


434 


106669 


58 


3 


789827 


269 


896236 


165 


893591 


434 


106409 


57 


4 


789988 


269 


896137 


165 


893851 


434 


106149 


56 


5 


790149 


269 


896038 


165 


8941 1 1 


434 


105889 


55 


6 


790310 


268 


895939 


165 


894371 


434 


105629 


54 


7 


790471 


268 


895840 


J65 


894632 


433 


105368 


53 


8 


790632 


268 


895741 


165 


894892 


433 


105108 


52 


9 


790', \fi 


268 


89;>641 


165 


8951,52 


433 


104848 


51 


10 


790954 


268 


895542 


165 


895412 


433 


104588 


50 


II 


9-791115 


268 


9-895443 


166 


9-895672 


433 


10-104328 


49 


1-2 


791-275 


i;<i7 


895343 


J 66 


895932 


433 


104068 


48 


jn 


791436 


2<.7 


895244 


166 


896192 


433 


103808 


47 


J4 


791596 


267 


895145 


11)6 


896452 


433 


103548 


46 


JT) 


791757 


267 


895045 


]f)6 


896712 


433 


103288 


45 


JH 


791917 


267 


894945 


1156 


896971 


433 


103029 


44 


J7 


792077 


267 


894846 


166 


897231 


433 


102769 


43 


1« 


792237 


266 


894746 


166 


897491 


433 


102509 


42 


J9 


792397 


266 


894646 


166 


897751 


433 


102249 


41 


20 


792557 


266 


894546 


166 


8980 JO 


433 


101990 


40 


21 


9-792716 


266 


9-894446 


167 


9-898270 


433 


10-101730 


39 


22 


792876 


266 


894346 


167 


898530 


433 


101470 


38 


23 


793035 


266 


894246 


167 


89H7H9 


433 


101211 


37 


24 


793195 


265 


894146 


167 


899(M9 


432 


100951 


36 


25 


793354 


265 


894046 


167 


899H(l8 


432 


100692 


35 


26 


793514 


265 


893946 


167 


899.568 


432 


100432 


34 


27 


793673 


265 


893846 


167 


899827 


432 


100173 


33 


28 


793832 


265 


893745 


167 


900086 


432 


099914 


.32 


29 


793991 


265 


893645 


167 


900346 


432 


099654 


31 


30 


794150 


264 


893544 


167 


900605 


432 


099395 


30 


3J 


9-794308 


264 


9-893444 


168 


9-900864 


432 


10099136 


29 


32 


794407 


264 


893343 


168 


901124 


432 


098876 


28 


33 


794626 


264 


893243 


168 


901383 


432 


098617 


27 


34 


794784 


264 


893142 


168 


901642 


432 


098358 


26 


3^ 


':'94942 


264 


893041 


168 


901901 


432 


098099 


25 


36 


795101 


264 


892940 


168 


902160 


432 


097840 


24 


37 


795259 


264 


892839 


168 


902419 


432 


097,581 


23 


38 


795417 


263 


892739 


168 


902679 


432 


097321 


22 


39 


795575 


263 


892638 


168 


902938 


432 


097062 


21 


40 


795733 


263 


892536 


168 


903197 


431 


096803 


20 


41 


9-795891 


263 


9-892435 


169 


9-9034,55 


431 


10096545 


19 


42 


796049 


263 


892334 


169 


903714 


431 


096286 


18 


43 


796206 


263 


892233 


169 


903973 


431 


096027 


17 


M 


796364 


262 


892132 


169 


904232 


431 


095768 


16 


45 


796521 


262 


892030 


169 


904491 


431 


095509 


15 


4G 


796679 


262 


891 929' 


169 


904750 


431 


0952.50 


14 


47 


796836 


262 


891827 


169 


905008 


431 


094992 


13 


48 


796993 


262 


891726 


169 


905267 


431 


094733 


12 


49 


797150 


261 


891624 


169 


9().5.V26 


431 


094474 


11 


50 


797307 


261 


891523 


170 


9(1.5784 


431 


094216 


10 


5] 


9-797464 


261 


9-891421 


no 


9-906043 


431 


10-0939.57 


9 


52 


797621 


261 


891319 


170 


9(t(>3()2 


431 


093698 


8 


53 


797777 


261 


891217 


170 


906.560 


431 


093440 


7 


54 


797934 


261 


891115 


170 


9068 19 


431 


093181 


6 


55 


798091 


261 


89 1013 


170 


907077 


431 


092923 


5 


56 


79H247 


261 


89(1911 


170 


907.3.36 


431 


092664 


4 


57 


79M403 


260 


890809 


no 


907.594 


431 


092406 


3 


58 


79H5H() 


260 


890707 


170 


907852 


431 


092148 


2 


59 


798716 


260 


890605 ■ 


170 


908111 


430 


091889 . 


1 


89 


798872 


260 


890503 


170 


908369 


430 


091631 






I Cosine | 



I Cotang. I 



Tang. 



61 Degrees. 



LOGARITHMIC SINES, COSINES, ETC. (39 Degrees. ) 



M. 


S.ne 


D. 


Cosine 


n. 


T.n-. 


D. 


Cot.n:.?. ; 







9-798872 


260 


9-89U503 


170 


9-9083G9 


430 


10-091031 


60 


1 


799U28 


260 


890400 


171 


908628 


430 


091372 


59 


2 


7i)9184 


260 


890298 


171 


90888G 


430 


091 1 14 


58 


3 


799339 


259 


890195 


171 


909144 


430 


09035G 


57 


4 


799495 


259 


890093 


171 


909402 


430 


090598 


50 


5 


799651 


259 


889990 


171 


909060 


430 


090340 


55 


(') 


7D980G 


259 


889888 


171 


909918 


430 


090082 


54 


7 


799962 


259 


889785 


171 


910177 


4.10 


C89823 


53 


8 


8:)01 17 


259 


889682 


171 


910435 


430 


0895G5 


52 


» 


800272 


258 


889579 


171 


910693 


430 


089307 


51 


10 


800427 


258 


889477 


171 


910951 


430 


089049 


50 


11 


9-800582 


258 


9-889374 


172 


9-911209 


430 


10-088791 


49 


12 


800737 


258 


889271 


172 


911467 


430 


088533 


48 


13 


800892 


258 


889168 


172 


911724 


430 


088276 


47 


14 


801047 


258 


889064 


172 


911982 


430 


088018 


46 


\:^ 


801201 


258 


888961 


172 


912240 


430 


0877GO 


45 


IG 


801356 


257 


888858 


172 


912498 


430 


087502 


44 


17 


801511 


257 


888755 


172 


91-2756 


430 


087244 


43 


l^ 


801065 


257 


888651 


172 


913014 


429 


086980 


42 


19 


801819 


257 


888548 


172 


913271 


429 


086729 


41 


2U 


801973 


257 


888444 


173 


913529 


423 


080471 


40 


21 


9-802 J 28 


257 


9-888341 


173 


9-913787 


429 


10-086213 


39 


22 


802-282 


250 


888^7 


173 


914044 


429 


085956 


38 


23 


802436 


256 


888134 


173 


914302 


429 


085098 


37 


24 


802589 


256 


888030 


173 


9145G0 


429 


085440 


36 


25 


8;;2743 


256 


887926 


173 


914817 


429 


085183 


35 


26 


802897 


256 


887822 


173 


915075 


429 


084925 


34 


27 


803050 


256 


887718 


173 


915332 


429 


084668 


33 


28 


803204 


256 


887614 


173 


915590 


429 


084410 


32 


29 


803357 


255 


887510 


173 


915847 


429 


084153 


31 


30 


803511 


255 


887400 


174 


916104 


429 


083896 


30 


31 


9-803664 


255 


9-887302 


174 


9-916362 


429 


10-083638 


29 


32 


803817 


255 


887198 


174 


916619 


429 


083381 


28 


33 


803970 


255 


887093 


174 


916877 


429 


083123 


27 


34 


804123 


255 


886989 


174 


917134 


429 


C82866 


26 


35 


804276 


254 


886885 


174 


917391 


429 


082609 


25 


36 


804428 


254 


886780 


174 


917648 


429 


082352 


24 


37 


804581 


254 


886676 


174 


917905 


429 


082095 


23 


38 


804734 


254 


886571 


174 


918163 


428 


081837 


22 


39 


804886 


254 


886466 


174 


918420 


428 


081580 


21 


49 


805039 


254 


886362 


175 


918677 


428 


081323 


20 


41 


9-805191 


254 


9-886257 


175 


9-918934 


428 


10-081066 


19 


42 


805343 


253 


886152 


175 


919191 


428 


0808)9 


18 


43 


805495 


253 


886047 


175 


919448 


428 


080552 


17 


44 


805647 


253. 


885942 


175 


919705 


428 


080295 


16 


45 


805799 


253 


885837 


175 


919902 


428 


080038 


15 


46 


805951 


253 


885732 


175 


920219 


428 


079781 


14 


47 


806103 


253 


885627 


175 


920476 


428 


079524 


13 


48 


806254 


253 


885522 


175 


920733 


428 


079207 


13 


49 


806406 


252 


885416 


175 


920990 


428 


079010 


11 


5U 


806557 


252 


885311 


176 


921247 


428 


078753 


10 


51 


9-806709 


252 


9-885205 


176 


9-921.503 


428 


10-078497 


9 


52 


806860 


252 


885100 


176 


921700 


428 


078-240 


8 


53 


807011 


252 


884994 


176 


922017 


428 


077983 


7 


54 


807163 


252 


884889 


176 


922274 


428 


077726 


6 


55 


807314 


252 


884783 


176 


922530 


428 


077470 


5 


56 


807465 


251 


884677 


176 


922787 


428 


077213 


4 


57 


807015 


251 


884572 


176 


923044 


428 


076956 


3 


58 


807766 


251 


884466 


176 


923300 


428 


076700 


2 


59 


807917 


251 


884360 


176 


923557 


427 


876443 


1 


60 


808067 


251 


884254 


177 


923813 


427 


076187 






I Sine I 



^ Couuag. \ 



I Tang. I M. 



^J>«^re» 



238 


(40 Deg 


recs.) 


LOGARITHMIC 


SIKES, COSiyFS, ETC. 




M. 1 


Sine 1 


D. 


Cosine 1 


D. 1 


Tang. 1 


D. 1 


Cotan?. 1 







9-«08(l67 


251 


9-884254 


177 


9-923813 


427 


W076 187^7 


60 


1 


808218 


251 


884148 


177 


924070 


427 


075y;«) 


59 


2 


808368 


251 


884042 


177 


924327 


427 


07.5673 


58 


3 


808519 


250 


883936 


177 


924583 


427 


075417 


57 


4 


8U8669 


250 


883829 


177 


924840 


427 


075160 


56 


5 


808819 


250 


883723 


177 


92.5096 


427 


' 074904 


55 


6 


808969 


250 


883617 


177 


925352 


427 


074648 


54 




809119 


250 


883510 


177 


925609 


427 


074391 


53 


H 


809269 


250 


883404 


177 


925865 


427 


074135 


52 


it 


809419 


249 


883297 


178 


926122 


427 


073878 


51 


10 


809569 


249 


883191 


178 


9-J6378 


427 


073622 


50 


11 


9-809718 


249 


9-883084 


178 


9-926634 


427 


10-073366 


49 


J -2 


809868 


249 


882977 


.178 


926890 


427 


073110 


48 


13 


8)0017 


249 


882871 


178 


927147 


427 


072853 


47 


14 


810167 


249 


882764 


178 


927403 


427 


072597 


46 


15 


810316 


'248 


882657 


178 


927659 


427 


072341 


45 


16 


8)0465 


248 


882550 


178 


927915 


427 


072085 


44 


IT 


810614 


248 


882443 


178 


928171 


427 


071829 


43 


J8 


810763 


248 


882336 


179 


928427 


427 


071573 


42 


J9 


810912 


248 


882229 


179 


928683 


427 


071317 


41 


20 


81 J 061 


248 


882121 


179 


928940 


427 


071060 


40 


21 


9-81 1210 


248 


9-882014 


179 


9-929196 


427 


10-070804 


39 


22 


811358 


247 


881907 


179 


929452 


427 


070548 


38 


23 


811507 


247 


881799 


179 


929708 


427 


070292 


37 


24 


811655 


247 


861692 


179 


929964 


426 


070036 


36 


25 


811804 


247 


881584 


179 


930220 


426 


069780 


35 


26 


811952 


247 


881477 


179 


930475 


426 


069525 


34 


27 


812100 


247 


881369 


179 


930731 


426 


069269 


33 


58 


8 12248 


247 


881201 


180 


930987 


426 


069013 


32 


29 


812396 


246 


881153 


180 


931243 


426 


008757 


31 


30 


812544 


246 


881046 


180 


931499 


426 


068501 


30 


31^ 


9-812692 


246 


9-880938 


180 


9-931755 


426 


10-068245 


29 


32 


812840 


246 


880830 


180 


932010 


426 


007990 


28 


33 


812988 


246 


880722 


180 


932266 


426 


0G7734 


27 


34 


813135 


246 


880613 


180 


932522 


426 


0G7478 


26 


35 


813283 


246 


880505 


180 


932V'/8 


426 


GG7222 


25 


36 


813430 


245 


880397 


180 


933033 


426 


0C69G7 


24 


37 


813578 


245 


880289 


181 


933289 


426 


066711 


23 


38 


813725 


245 


880180 


181 


933545 


426 


066455 


22 


39 


813872 


245 


880072 


181 


933800 


426 


066200 


21 


40 


814019 


245 


879963 


181 


934056 


426 


065944 


20 


41 


9-814166 


245 


9-879855 


181 


9-934311 


426 


10-065689 


19 


42 


814313 


245 


879746 


18/ 


9345()7 


426 


065433 


18 


43 


814460 


244 


879637 


181 


934823 


426 


065177 


17 


44 


814607 


244 


879529 


181 


935078 


426 


064922 


16 


45 


814753 


244 


879420 


181 


935333 


426 


0646r)7 


15 


46 


814900 


244 


879311 


181 


935589 


426 


064411 


14 


47 


815046 


244 


879202 


182 


935844 


426 


064156 


13 


48 


815193 


244 


879093 


182 


936100 


426 


0f;3900 


12 


49 


815339 


244 


878984 


182 


936355 


426 


0(53645 


11 


50 


815485 


243 


878875 


182 


936610 


426 


063390 


10 


51 


9-8i.v;3i 


243 


9-8787(16 


182 


9-936866 


425 


10-0631.34 


9 


52 


815778 


243 


878656 


]H2 


937121 


425 


062879 


8 


53 


815924 


243 


878547 


182 


937376 


425 


062624 


7 


54 


8lti069 


243 


878438 


182 


9376.32 


425 


062.3<)8 


6 


55 


816215 


243 


878328 


182 


937887 


425 


062)13 


5 


56 


816361 


243 


878219 


183 


938142 


425 


0618.58 


4 


57 


8:r,5(»7 


242 


878109 


183 


938398 


425 


001602 


3 


58 


816652 


242 


877999 


183 


938653 


425 


061347 


2 


59 


816798 


242 


877890 


183 


938908 


425 


061092 


1 


60 


816943 


242 


877780 


183 


939163 


425 


060837 






I Coauie i 



Sina ) 



I Cotang. J 



Tang. 1 



.^Df^gnies. 



LOGARITHMIC iSIXES, fV SINES, ETC. ^41 Degrees.) 23? 



M. 


Sine 


D. 


Cosine 


1). 


Tang. 


D. 


Cotang. 







9-816943 


242 


9 877780 


183 


9-939163 


425 


10-060837 


60 


1 


817088 


242 


877ti70 


183 


939418 


425 


060582 


50 


2 


817233 


242 


W77.3f;0 


183 


939673 


425 


060327 


58 


3 


817379 


242 


8774.50 


183 


939928 


425 


060072 


57 


4 


817524 


241 


877;{40 


183 


940Jf33 


425 


059817 


56 


5 


817668 


241 


877230 


184 


940438 


425 


059562 


53 


6 


817813 


241 


877120 


184 


940094 


4-.;5 


039306 


54 


7 


817958 


241 


877010 


184 


940949 


423 


059051 


33 


8 


818103 


241 


876899 


1^<4 


941204 


425 


058796 


32 


9 


81.-^*47 


241 


876789 


184 


941438 


425 


058542 


31 


10 


81831)2 


241 


876678 


184 


941714 


425 


058286 


50 


11 


9-81H536 


240 


9 876368 


184 


9-94 19G8 


425 


100.')8032 


49 


12 


818681 


240 


87()457 


184 


942223 


. 425 


037777 


48 


i:j 


818825 


240 


871)347 


184 


942478 


425 


037.322 


47 


14 


818969 


240 


87ti236 


185 


942733 


425 


057267 


46 


J 5 


819113 


240 


876125 


185 


i 942988 


425 


■ 0.37(112 


45 


16 


819257 


240 


876014 


185 


943243 


425 


05(:7.-,7 


44 


17 


8194.11 


240 


875904 


185 


943498 


425 


05(_.3l/2 


43 


18 


819.145 


239 


873793 


183 


9437 .")2 


425 


056248 


42 


19 


8iyt;89 


239 


873682 


185 


944(107 


425 


055993 


41 


20 


81ii832 


239 


875571 


185 


944262 


425 


055738 


40 


21 


9-819976 


239 


9-875439 


185 


9-944317 


425 


10-05.5483 


39 


22 


82012(1 


239 


873348 


185 


944771 


424 


053229 


38 


23 


82021 >3 


239 


87.->237 


185 


943026 


424 


054974 


37 


'>4 


821)406 


239 


873126 


186 


945281 


424 


034719 


36 


^5 


82()5.")0 


238 


873014 


186 


945335 


424 


054465 


35 


% 


8206!)3 


238 


874903 


186 


945790 


424 


054210 


34 


27 


820836 


238 


874791 


18() 


946045 


424 


053955 


33 


28 


820979 


238 


874680 


186 


94(5299 


424 


0.33701 


32 


29 


821122 


238 


874568 


186 


94(]554 


424 


05344(j 


31 


30 


821265 


238 


874436 


186 


946808 


424 


053192 


3C 


31 


9-821407 


238 


9-874344 


186 


9-947063 


424 


10052937 


m 


132 


821530 


238 


874232 


187 


947318 


424 


052682 


28 


33 


8210U3 


237 


874121 


187 


947572 


424 


052428 


27 


."■1 


821835 


237 


874009 


187 


947826 


424 


052174 


2f 


33 


821977 


237 


873896 


187 


948(181 


424 


051919 


25 


36 


822120 


237 


873784 


187 


948.136 


424 


051664 


24 


37 


82-22:32 


237 


873G72 


187 


948390 


424 


051410 


23 


38 


82241)4 


237 


8733C0 


187 


948844 


424 


051156 


22 


39 


822546 


237 


87.3448 


187 


949099 


424 


050901 


21 


40 


822688 


236 


873335 


187 


949333 


424 


050647 


20 


41 


9-822830 


236 


9-873223 


187 


9-949607 


424 


10.050.393 


19 


42 


822972 


236 


873110 


188 


949862 


424 


050138 


18 


43 


823114 


236 


872998 


188 


950116 


424 


049884 


17 


44 


823255 


236 


872885 


1P8 


950370 


424 


049630 


16 


45 


823397 


236 


872772 


IHR 


950625 


424 


049375 


15 


46 


823339 


2.36 


872639 


188 


950879 


424 


049121 


14 


47 


823HS0 


235 


872.^47 


188 


951133 


424 


048867 


13 


48 


823Hi>l 


235 


87LM-4 


188 


931388 


424 


048612 


12 


49 


8239(i3 


235 


8723.21 


188 


95ir'»2 


424 


048338 


11 


50 


824104 


235 


872208 


188 


95189<) 


424 


048104 


10 


51 


9-824245 


235 


9-872093 


189 


9-9.32150 


424 


10-047850 


9 


52 


8243H6 


235 


871981 


189 


9.32405 


424 


047595 


a 


53 


824.V27 


235 


871H68 


189 


952659 


424 


047341 


7 


54 


82461 « 


234 


8717.35 


189 


9.52913 


424 


047087 


6 


55 


824808 


234 


871641 


189 


9.33167 


423 


046833 


5 


56 


824949 


234 


871528 


189 


9.33421 


423 


046579 


4 


57 


823090 


234 


871414 


189 


9.-)3675 


423 


046325 


3 


58 


82.3230 


2.34 


871301 


189 


933929 


423 


046071 


<r 


59 


82.3371 


234 


871187 


189 


9.54183 


423 


045817 


1 


60 


825511 


234 


871073 


190 


954437 


423 


045563 






I Cosine I 



J Sine 



Cotang. I 



I Tang. 



48 Degrees. 



24 > (4li JJogiccs.) LOiJAniTinilC SINES, COSINES, ETC. 



M. 


Sine 


1 D. 


Cos.ne 


D. 


Ta„g. 


1 D. 


1 Cotang. 


1 





9-825511 


234 


9-871073 


190 


9-954437 


423 


10-045563 


60 


I 


823(i51 


233 


870960 


190 


954691 


423 


0453(t9 


59 


2 


825791 


233 


870846 


190 


954945 


423 


045055 


58 


3 


825^31 


233 


870732 


190 


955200 


423 


044800 


57 


4 


82607 1 


233 


870618 


190 


955454 


423 


044546 


56 


5 


8262 11 


233 


870504 


190 


955707 


423 


044293 


55 


6 


826351 


233 


870390 


190 


955961 


423 


044039 


54 


7 


826491 


233 


870276 


190 


956215 


423 


043785 


53 


8 


826631 


233 


870161 


190 


956469 


423 


043531 


52 


9 


826770 


232 


870047 


191 


956723 


423 


043277 


51 


10 


826910 


232 


869933 


191 


956977 


423 


043023 


50 


11 


9-827049 


232 


9-869818 


191 


9-957231 


423 


10-042769 


49 


V2 


827189 


232 


869704 


191 


957485 


423 


042515 


48 


i:t 


827328 


232 


869589 


191 


957739 


423 


042261 


47 


14 


827467 


232 


869474 


191 


957993 


423 


042(»07 


46 


15 


827606 


232 


8693C0 


191 


958246 


423 


041754 


45 


16 


827745 


232 


869245 


191 


958500 


423 


04 1500 


44 


17 


■ 827884 


231 


869:30 


191 


958754 


423 


041246 


43 


18 


828023 


231 


869015 


192 


959008 


423 


04(<992 


42 


19 


828162 


231 


868900 


192 


9592(52 


423 


040738 


41 


20 


828301 


231 


868785 


192 


959516 


423 


040484 


40 


2] 


9-828439 


231 


9-86P670 


192 


9-959769 


423 


10-040231 


39 


22 


828578 


231 


8()8555 


192 


960023 


423 


039977 


;i8 


23 


828716 


231 


8()8440 


192 


900277 


423 


039723 


37 


24 


828855 


230 


8()8324 


192 


900531 


423 


039469 


36 


25 


828993 


230 


8(i8209 


192 


960784 


423 


039216 


35 


26 


829131 


230 


868093 


192 


961038 


423 


02C962 


34 


27 


829269 


230 


867978 


193 


961291 


423 


Gr07C9 


33 


28 


829407 


230 


86:8i;2 


193 


901545 


423 


occ-:c5 


52 


29 


829545 


230 


867747 


193 


901799 


423 


C38201 


31 


30 


829683 


230 


867631 


193 


962U52 


423 


037C48 


30 


31 


9-829821 


229 


9-867515 


193 


9-962306 


423 


10-C37C94 


29 


32 


829959 


229 


867;!:i9 


193 


9C25C0 


423 


037440 


28 


33 


830097 


229 


8(J7283 


193 


9C2813 


423 


037187 


27 


34 


830234 


229 


867167 


193 


963067 


423 


036933 


26 


35 


830372 


229 


867051 


193 


903320 


423 


036680 


25 


3t) 


830509 


229 


86(3935 


194 


963574 


423 


036426 


24 


37 


830646 


229 


866819 


194 


903827 


423 


036173 


23 


38 


830784 


229 


866703 


194 


9G4081 


423 


035919 


22 


39 


830921 


228 


8<)(5586 


194 


964335 


423 


035665 


21 


40 


831058 


228 


866470 


194 


964588 


422 


035412 


20 


41 


9-831195 


228 


9-866353 


194 


9-964842 


422 


10035158 


19 


42 


831332 


228 


806237 


194 


965095 


422 


034905 


18 


43 


831469 


228 


866120 


194 


965349 


422 


034651 


17 


44 


831606 


228 


866004 


195 


965602 


422 


034398 


16 


45 


831742 


228 


865887 


195 


965855 


422 


034145 


16 


46 


831879 


228 


8P5770 


195 


966109 


422 


033891 


14 


47 


832015 


227 


865653 


195 


9()63()2 


422 


033638 


13 


48 


832152 


227 


80.':536 


195 


966616 


422 


033384 


12 


49 


832288 


227 


865419 


195 


966869 


422 


033131 


11 


50 


832425 


227 


865302 


195 


967123 


422 


032877 


10 


51 


9-832561 


227 


9'865185 


195 


9-967376 


422 


10032624 


9 


52 


832697 


227 


865068 


195 


967629 


422 


032371 


8 


53 


832833 


227 


864950 


195 


967883 


422 


032117 


7 


54 


832969 


226 


864833 


196 


968136 


422 


03IIS64 


6 


55 


833105 


226 


864716 


196 


968389 


422 


031611 


5 


56 


833241 


226 


864598 


196 


968643 


422 


031357 


4 


57 


833377 


226 


864481 


196 


968896 


422 


031104 


3 


58 


833512 


226 


864363 


196 


969149 


422 


030851 


2 


59 


833648 


226 


864245 


196 


969403 


422 


030597 


1 


60 


833783 


226 


864127 


196 


969656 


422 


030344 






Cotang. ^ 



Tang. 



47 Degrees. 



LOGARITHMIC SINES, COSINES, ETC. (43 Degrees.) 241 



I D. I Cosii 



I Cotaii^, 






9-833783 


226 


9-864127 


196 


9-9G9656 


4-.-2 


10-030344 


1 60 


1 


8339 J 9 


225 


864010 


196 


9G9909 


422 


030091 


59 


2 


834054 


225 


863892 


197 


970 162 


422 


029838 


58 


3 


834189 


225 


863774 


197 


970416 


422 


029584 


57 


4 


834325 


225 


863656 


197 


970669 


422 


029331 


56 


5 


8344G0 


225 


863538 


197 


970922 


422 


029078 


55 


6 


834595 


2-25 


863419 


197 


971175 


422 


028825 


54 


7 


834730 


225 


863301 


197 


971429 


422 


028571 


53 


& 


8348G5 


225 


863183 


197 


971682 


422 


028318 


52 


9 


834999 


224 


8630G4 


197 


971935 


422 


028(.C5 


51 


10 


835134 


2-24 


8G2946 


198 


972 188 


422 


027812 


50 


11 


9-835269 


224 


9-862827 


198 


9-972441 


422 


10-027559 


49 


12 


835403 


224 


862709 


iJ8 


972694 


422 


027306 


48 


13 


835538 


224 


862590 


198 


972948 


422 


027052 


47 


14 


835672 


224 


86247] 


198 


973201 


4-22 


G26799 




15 


835807 


224 


862353 


198 


973454 


422 


C2G54G 


45 


IG 


835941 


224 


862234 


198 


973707 


422 


026293 


44 


17 


&3G075 


223 


8C2115 


198 


973960 


422 


02G040 


43 


18 


83G209 


223 


8G199G 


198 


974213 


4-22 


025787 


4 -J 


19 


836343 


223 


861877 


198 


9744G6 


422 


C25534 


41 


20 


83G477 


223 


8G1758 


199 


974719 


422 


025281 


40 


21 


9-836611 


223 


9-861C38 


199 


9-974973 


4-22 


10-025027 


39 


22 


836745 


223 


8GI5J9 


199 


97522G 


422 


02477-1 


38 


23 


836878 


223 


8G1400 


1D9 


975479 


422 


(;:4.:2i 




24 


837012 


222 


861280 


199 


975732 


422 


024268 


36 


25 


837146 


222 


8GIIG1 


1J9 


9751.85 


422 


G24015 


35 


2G 


S37279 


222 


8G1U41 


199 


97G238 


422 


0237C2 


34 


27 


837412 


222 


860922 


199 


97G491 


422 


0235G9 


33 


28 


837546 


ooo 


860802 


199 


97G744 


422 


023256 


.'52 


29 


837679 


222 


860682 


200 


97G997 


422 


023003 


31 


30 


837812 


222 


8;;o562 


200 


977250 


422 


C22750 


30 


31 


9-837945 


222 


9-860442 


200 


9-977503 


422 


10-02-2497 


29 


32 


838078 


221 


8G0322 


200 


977756 


422 


022244 


28 


33 


838211 


221 


860202 


200 


978009 


422 


02 J 991 


27 


34 


838344 


221 


8G0082 


200 


9782G2 


422 


021738 


2() 


35 


838477 


221 


8599G2 


200 


978515 


422 


021485 


25 


3G 


838G10 


221 


859842 


200 


97t<7G8 


422 


021232 


24 


37 


838742 


221 


859721 


201 


!y7'JU21 


422 


02U979 


23 


38 


838875 


221 


859601 


201 


t)'7<l274 


422 


02072G 


22 


39 


839007 


221 


859480 


201 


97-j.;27 


422 


020473 


21 


4U 


839140 


220 


859360 


201 


97'J780 


422 


020220 


20 


41 


9-839272 


220 


9-859239 


201 


9-980033 


422 


10-0 J 9967 


19 


42 


839404 


220 


8591 1<J 


201 


98028(5 


422 


019714 


18 


43 


839536 


220 


858998 


201 


980538 


422 


019402 


17 


44 


8396tJ8 


220 


858877 


201 


980791 


421 


019209 


16 


45 


830800 


220 


858756 


202 


981044 


421 


018956 


15 


4G 


839932 


220 


858635 


202 


981297 


421 


018703 


14 


47 


8400G4 


219 


858514 


202 


981550 


421 


018450 


13 


48 


840196 


219 


858393 


202 


981803 


421 


018197 


n 


49 


840328 


219 


858272 


202 


982U56 


421 


017944 


11 


50 


840459 


219 


858151 


202 


982309 


421 


017691 


10 


51 


9-840591 


219 


9-858029 


202 


9-982562 


421 


10-017438 


9 


52 


840722 


219 


857908 


202 


982814 


421 


017186 


8 


53 


840854 


219 


857786 


202 


983i)G7 


421 


016933 


7 


54 


840985 


219 


857665 


203 


983320 


421 


016G80 


6 


55 


841116 


218 


857543 


203 


983573 


421 


016127 


5 


5G 


841247 


218 


857422 


203 


983826 


421 


016174 


4 


57 


841378 


218 


857300 


203 


984079 


421 


015921 


3 


58 


841509 


218 


357178 


203 


984331 


421 


015669 


2 


59 


841640 


218 


857056 


203 


984584 


421 


015416 


1 


60 


841771 


218 


856934 


203 


984837 


421 


015163 






J C<wne I 



I I Cotang. I I 

46 Degrees. 



Tang. I ai 



242 (44 Degrees.) LOGABTTIIMIC SINES, COSINES, ETC 



M. 


1 Sine 
9-841771 


Q 


1 Cosine 


1 D. 


1 Tan?. 


1 D. 


I Cotang. 


1 





•218 


9-856934 


203 


y-984837 


421 


10-015163 


60 


1 


841902 


218 


85G812 


203 


985090 


421 


014910 


59 


2 


842033 


218 


856690 


204 


985343 


421 


014657 


58 


3 


842163 


217 


856568 


204 


985596 


421 


014404 


57 


4 


842294 


217 


856446 


204 


985848 


4J1 


014152 


36 


5 


842424 


217 


85G323 


204 


980101 


421 


013899 


55 


'J 


842555 


217 


856201 


204 


986354 


421 


013646 


54 


7 


842685 


217 


856078 


204 


986607 


421 


013393 


53 


8 


842815 


217 


855956 


204 


986860 


421 


013140 


52 


9 


842946 


217 


855833 


204 


987112 


421 


012888 


51 


10 


843076 


217 


855711 


205 


987305 


421 


012635 


50 


11 


9-843206 


216 


9-855588 


205 


9-987618 


421 


10-012382 


49 


n 


843336 


216 


855465 


205 


987871 


421 


012129 


48 


13 


843466 


216 


855342 


205 


988123 


421 


011877 


47 


14 


843595 


216 


855219 


205 


988376 


421 


«11624 


46 


15 


843725 


216 


855096 


205 


988G29 


421 


011371 


45 


16 


843855 


216 


854973 


205 


988882 


421 


011118 


44 


17 


843984 


216 


854850 


205 


989134 


421 


010866 


43 


18 


844J14 


215 


854727 


206 


989387 


421 


010613 


42 


19 


844243 


215 


854003 


206 


989640 


421 


010360 


41 


20 


844372 


215 


854480 


206 


989893 


421 


010107 


40 


21 


9-844502 


215 


9-854356 


206 


9-990145 


421 


10-009855 


39 


22 


844G31 


215 


854233 


206 


990398 


421 


009602 


38 


23 


8447G0 


215 


854109 


206 


990G51 


421 


009349 


37 


24 


844889 


215 


853986 


206 


990903 


421 


009007 


36 


25 


845018 


215 


8538(52 


206 


991 156 


421 


008844 


35 


26 


845147 


215 


853738 


206 


991409 


421 


008591 


34 


27 


845276 


214 


8536 14 


207 


99 1662 


421 


008338 


33 


28 


845405 


214 


853490 


207 


991914 


421 


008086 


32 


29 


845533 


214 


8533(56 


207 


992 167 


421 


007833 


31 


30 


845662 


214 


853242 


207 


992420 


421 


007580 


30 


31 


9-845790 


214 


9-853118 


207 


9-992672 


421 


10-007328 


29 


32 


845919 


214 


852994 


207 


992925 


421 


007075 


28 


33 


846047 


214 


852869 


207 


993178 


421 


006822 


27 


34 


846175 


214 


852745 


207 


993430 


421 


006570 


26 


35 


846304 


214 


852620 


207 


993683 


421 


006317 


25 


36 


846432 


213 


852496 


208 


993930 


421 


006064 


24 


37 


846560 


213 


852371 


208 


994189 


421 


005811 


23 


38 


846688 


213 


852247 


208 


994441 


421 


005559 


22 


3i) 


846816 


213 


852122 


2U8 


994694 


421 


005306 


21 


40 


846944 


213 


851997 


208 


994947 


421 


005053 


20 


41 


9-847071 


213 


9-851872 


208 


9-995199 


421 


10004801 


19 


12 


847199 


213 


851747 


208 


995452 


421 


004548 


18 


43 


847327 


213 


851622 


208 


995705 


421 


004295 


17 


44 


847454 


212 


851497 


209 


995957 


421 


004043 


16 


45 


847582 


212 


851372 


209 


996210 


421 


003790 


15 


46 


847709 


212 


851246 


209 


996463 


421 


003537 


14 


47 


847836 


212 


851121 


209 


996715 


421 


003285 


13 


48 


847964 


212 


850996 


209 


996968 


421 


003032 


12 


49 


848091 


212 


850870 


209 


997221 


421 


002779 


11 


50 


848218 


212 


850745 


209 


997473 


421 


002527 


10 


51 


9-848345 


212 


9-850619 


209 


9-997726 


421 


10002274 


9 


52 


848472 


211 


850493 


210 


997979 


421 


002021 


8 


53 


848599 


211 


850368 


210 


998231 


421 


001769 


7 


54 


848726 


211 


850242 


210 


99S484 


421 


001516 


6 


55 


848852 


211 


850116 


210 


998737 


421 


001263 


5 


5(i 


848979 


211 


849990 


210 


998989 


421 


001011 


4 


57 


849106 


211 


849864 


210 


999242 


421 


000758 


3 


58 


849232 


211 


849738 


210 


999495 


421 


000505 


2 


59 


849359 


211 


849611 


210 


999748 


421 


000253 


1 


60 


849485 


211 


849485 


210 


10-000000 


421 


000000 







Cosine 




Sine 

4 


1 
5 Pegre« 


Cotang. 




Tang. 1 


M. 



TABLE XIV. 
IS'ATURAL SIGNS AND TAl^GENTS^ 



244 



NATURAL SINES. 



0° ' 1° i 2° 


3° i 


4° 


5° 


G° 


7° 


/ 


•000 0000 -017 452t|-034 899o'-052 3360'-069 7565 


•087 1557 


•104 5285 


121 8693 


60 


2909, 74321-0351902, 62G4:-070 0467 


4455 


8178 


1221581 


59 


5818-918 0341 4809' 9163! 3368 


7353 


•105 1070 


4168 


58 


87 -271 3249 7716 


•053 20741 6270 


•088 0251 


3963 


7355 


57 .; 


■0011G33! 6158!-0360C23 


4979; 9171 


3148 


6856 


123 0241 


56 ' 


4544' 906Gi 3530 


7383! •071 2073 


CO 10 


9748 


3128 


55 : 


7453-0191974 6437 


•054 0788j 4974 


891.3 


•106 2641 


G015 


54 


•00203G2; 48831 9344 


3633| 7876 


•0891S43 


5533 


8301 


53 


3-271 7731 '-037 2251 


6537 1-072 0777 


4733 


84C5 


1241788 


52 


G180;-020 0G33i 5158 


9502 


3378 


7u35 


•107 1313 


4674 


51 


9089! 360SJ 8065 


•055 2103 


6580 


•090 0532 


4210 


75C0 


50 


•0031938! 6510-038 0971 


5311 


9481 


3429 


7102 


125 0446 


49 


4307' 9424 3378 


8215 


-073 2382 


6323 


9934 


3332 


4S 


7815 '-021 2332 C7S5 


•0561113 


5233 


9223 


•108 2885 


6218 


47 


•004 07241 52111 9G92 


4024 


8184 


-091 2119 


5777 


9104 


46 


3633! 8143r03£2538 


6923 


•074 1085 


5016 


8C69 


126 1990 


45 


6542:-0221057 5505 


9832 


3986 


• 7913 


•ir,9 1560 


4875 


44 


9451 39651 8411 


■057 2733 


6887 


-0920809 


4452 


7761 


43 


■0052360 G873I-0401318 


5340 


9787 


3706 


7343 


127 0646 


42 


5268 97811 4224 


8541 


•075 2088 


6602 


-110 0234 


3531 


41 


8177 '•023 2690 7131 


•058 i448 


5589 


9499 


3126 


6410 


40 


•0061086i 55981-0410037 
3935 1 85061 2944 


4352 


8489 


-093 2395 


C017 


9302 


39 


7256 


-0761330 


5291 


8908 


•128 2186 


38 


69041-024 14141 5350 


•059 OIGO 


4230 


8187 


-111 1799 


5071 


37 


9813 4322J 8757 


3061 


7190 


-094 1083 


4689 


7950 


36 


-007 2721 72301-042 1663 


5967 


-077 0091 


3979 


7580 


•129 0841 


35 


5630-025 01381 4509 


8871 


2991 


6675 


•112 0471 


3725 


34 


8539 1 3046 7475 


•0601775 


5891 


9771 


3301 


6609 


33 


•008 1448/ 5954|-043 0382 


4078 


8791 


-095 2666 


6252 


9494 


32 


4357, 8862 3288 


7582 


-078 1091 


5562 


9142 


•130 2378 


31 


72651-0261769 6194 


-061 0485 


4591 


8458 


•113 2032 


5262 


30 


•009 0174 46771 9100 


3389 


7491 


-0961353 


4922 


8146 


29 


3083 7685,-044 2000 


6292 


-079 0391 


4248 


7812 


•131 1030 


23 


5992 -027 04931 4912 


9196 


3290 


7144 


■114 0702 


3913 


27 


89001 3401' 7818 


-062 2099 


6190 


-097 0039 


3532 


6797 


26 


•0101809 6309 ;^045 0724 


5002 


9090 


2934 


6482 


9681 


25 


4718 9-210: 3630 


7905 


-080 1989 


58-29 


9372 


•132 2564 


24 


76271-028 2124: 6536 


•0630808 


4889 


8724 


•115 2201 


5447 


23 


-01105351 5032 9442 


3711 


7788 


-098 1619 


5151 


8330 


22 


3444| 7940-046 2347 


6614 


-081 0687 


4514 


8010 


•1331213 


21 


6353!^029 0847i 5253 


9517 


3587 


7408 


■116 0929 


4096 


20 


9261 3755; 8159 


•064 2120 


6486 


-099 0303 


3818 


6979 


19 


-012 2170 6662 -047 1065 


5323 


9385 


3197 


6707 


9862 


18 


5079 9570, 3970 


■8226 


-082 2284 


6092 


9596 


•134 2744 


17 


7987-030 2478' 6870 


•065 1129 


5183 


8986 


■117 2485 


5627 


16 


•313 0S95J 5385' 9781 


4031 


8082 


-100 1881 


5374 


8509 


15 i 


3805 8293-048 2687 


6934 


-083 0981 


4775 


8263 


•135 1392 


14 


67131-0311-200, 5592 


9836 


3880 


7669 


•1181151 


4274 


13 


9622| 4108 8498 


-066 2739 


6778 


-101 0503 


4040 


7156 


12 


■014 -2530 7015-0491403 


5641 


9677 


3457 


6928 


■136 0038 


11 


5439! 9922 4308 


8544 


-084 2570 


6351 


9816 


2919 


10 


83481-032 2830 7214 


•067 1446 


5474 


9245 


-119 2704 


5801 


9 


-0151256' 5737-050 0119 


4349 


8373 


-102 2138 


5593 


8683 


8 


4165 8644 3024 


7-251 


•085 1271 


5032 


8481 


-137 1564 


. 7 


7073,-0331552 59-29 


-068 0153 


4169 


7925 


-1-201368 


4445 


6 


9982; 4459 8835 


3055 


7067 


-103 0819 


4256 


73-27 


5 


■016 •28901 7366 ■051 1740 


5957 


9966 


3712 


7144'-138 0208 


4 


5709/034 0274 4645 


8859 


•086 2864 


6605 


•1-21 0031 


3089 


3 


8707: 3181 7550 


•009 1761 


5762 


9499 


2919 


5070 


2 


-0171616. 6088-052 0455 


4663 


8660 


-104 2392 


5806 


8850' 1 


4524 8995 3360 


7565 


•087 1557 


5285 


8693 


•13917311 


8'J° 1 88° 1 87° 


86° 


85° 


84° 


.83° 


82° 1 >' 




NAT. C 


OSINE. 











NATURAL SINES. 



24ri 



/ 


8° 


9° 


10° 


11° 


12° 


13° 


14° 


1-:° 


/ 





1391731 


156 4345 


173 6482 


190 8090 


•207 9117 


2-24 9511 


241 9219 


258 8190 


CO 


i 1 


4G12 


721S 


93-lG 


191 0945 


•208 1902 


225 2345 


2 1:2 2041 


250 LXI 


5'J 


., 


7492 


157 0091 


174 2211 


3801 


4807 


5179 


4863 


3810 


58 


3 


140 0372 


29Gn 


5075 


6656 


7652 


8013 


7G85 


eeio 


57 


4 


3252 


583t 


7939 


9510 


•209 0497 


226 0846 


•243 0507 


9428 


56 


5 


G132 


8705 


175 0803 


•192 2365 


3341 


3680 


3329 


260 2-237 


55 


6 


9012 


158 1581 


3G67 


5220 


6186 


6513 


6150 


5045 


b\ 


7 1411892 


4453 


6531 


8074 


9030 


934G 


8971 


7853 


53 


8 ' 4772 


7325 


9395 


•193 0928 


•210 1874 


227 2179 


2441792 


261,0662 


52 


9| 


7651 


159 0197 


176 2258 


3782 


4718 


5012 


4G13 


3469 


51 


10 


142 0531 


3069 


6121 


6636 


7561 


7844 


7438 


6277 


50 


11 


341U 


5940 


7984 


9490 


•211 0405 


228 0677 


245 0254 


90S; 


4:) 


12 


6289 


8812 


177 0847 


•194 2344 


3248 


3509 


3074 


'262189^ 


48 


13 j 


91Gb 


•IGO 1GS3 


3710 


5197 


6091 


0341 


5891 


469; 


17 


14 1 


143 2047 


4555 


6573 


8050 


8934 


9172 


8713 


750 


40 


15 i 


492u 


7426 


9435 


•195 0903 


•2121777 


2-29 2004 


246 1533 


■203 031: 


15 


1(5 i 


7805 


•161 0297 


178 2298 


3756 


4619 


4835 


4352 


311' 


44 


17 ■•144 0684 


3167 


5160 


6609 


7462 


7666 


7171 


592; 


43 


IS 


3562 


6038 


8022 


9iCl 


•213 0304 


230 0497 


9990 


S73-; 


42 


19 


6440 


8909 


•179 0884 


•196 2314 


3146 


332S 


247 2809 


2G4153i 


41 


20 


9319 


•162 1779 


3746 


5166 


5988 


6159 


50'27 


4342 


40 


21 


•145 2197 


4650 


C607 


8018 


8829 


8989 


8445 


7147 


39 


22 


5075 


7520 


9469 


•197 0870 


•2141671 


231 1819 


248 1263 


9952 


38 


23 


7953 


•163 0390 


•180 2330 


3722 


4512 


4649 


4081 


265 2757 


37 


24 


•146 0830 


3260 


5191 


6573 


7353 


7479 


6899 


5561 


36 


25 


3708 


6129 


8052 


9425 


•215 0194 


232 0309 


9716 


8366 


35 


26 


6585 


8999 


-181 0913 


•198 2276, 30351 


3138 


•249 2533 


2661170 


34 


27 


9463 


•1641868 


3774 


5127 


5876 


5967 


5350 


3973 


33 


28 


•147 2340 


4738 


6635 


7978 


8716 


8796 


8167 


6777 


32 


2'J 


5217 


7607 


9495 


•199 0829 


•2161556 


•2331625 


■250 0984 


9581 


31 


30 


8094 


•165 0476 


■182 2355 


3679 


4396 


4454 


3800 


-267 2384 


30 


31 


•148 0971 


3345 


5215 


6530 


7236 


7282 


6616 


5187 


29 


32 


3848 


6214 


8075 


9380 


•217 0076 


•234 0110 


9432 


7989 


28 


33 


6724 


9082 


•183 0935 


•200 2230 


2915 


2938 


•251 2248 


•268 0792 


27 


34 


9601 


•1661951 


3795 


5080 


5754 


5706 


5063 


3594 


20 


35 


•149 2477 


4819 


6654 


7930 


8593 


8594 


7879 


6396 


25 


36 


5353 


7687 


9514 


•201 0779 


•218 1432 


•235 1421 


•252 0694 


■ 9198 


24 


37 


8230 


•167 0556 


■184 2373 


3629 


4271 


4248 


3508 


•2^9 2000 


23 


38 


•150 not 


3423 


5232 


6478 


7110 


7075 


63-23 


4801 


22 


J9 


3981 


6291 


8091 


9327 


9948 


9902 


9137 


7602 


21 


to 


6857 


9159 


•185 0949 


•202 2176 


•219 2786 


•236-2729 


•253 1952 


■270 0403 


20 


si ' 973:^ 


•168 2026 


3808 


5024 


5624 


5555 


4766 


3204 


19 


12 1 1512608 


4894 


6666 


7S73 


8462 


8381 


7579 


6004 


18 


43 


1 5484 


7761 


9524 


•203 0721 


•2201300 


•237 1207 


•2540393 


8805 


17 


44 


I 8359 


•169 0628 


•186 2382 


3569 


4137 


4033 


3206 


•271 1605 


16 


45 


1521234 


3495 


5240 


6418 


6974 


6859 


6019 


4404 


15 


46 


410'j 


G362 


8098 


9265 


9811 


9684 


8S32 


7204 


14 


47 


6984 


9-228 


•187 0956 


•204 2113 


•221 2648 


•238 2510 


•255 1645 


•2720003 


13 


48 


985S 


•170 2095 


3813 


4961 


5485 


5335 


4158 


2802 


12 


49 


■153 2732 


4961 


6670 


7808 


8321 


8159 


7270 


5601 


11 


50 


5607 


7828 


9528 


•205 065S 


•2221158 


•239 0984 


■256 0082 


8400 


10 


51 


848--; 


•171 0694 


•188 2385 


3502! 3994 


3808 


2894 


•2731198 


9 


52 


•1541351 


3560 


5241 


6349 6830 


6633 


5705 


3997 


8 


53 


423C 


6425 


8095 


91951 9666 


9457 


8517 


6794 


7 


54 


710-1 


9291 


•189 0954 •206 2042 -223 2501 


•240 2280 


•257 1328 


9592 


6 


55 


9975 


•172 215P 


3811 4888 5337 


5104 


4139 


•274 2390 


5 


56 


•155 285] 


5022 


6667 77341 8172 


79-27 


6950 


5187 


4 


57 


572e 


7887 


9523 -207 0580 -224 1007 


•241 0751 


976( 


79S4 


3 


58 


8595 


•173 0752 


■190 237C 


3426, 3842 


3574 


-258 -2570 


•275 0781 


2 


59 


•156 1471 


3:;i7 


523^ 


t 6272 6676 


G39C 


5381 


3577 


1 


60 


434^ 


) 648i 


809( 


) 9117 9511 


9219 


8190 


6374 





/ 


81° 


80° 


79° 


1 78° 77° 


76° 


75° 


74° 


/ 


; 








NAT. 


COSINE 











246 



NATURAL SINES. 



16° 


17^^ 1 


18° 


19° 


20° 


21° 


22° 1 


23° ! 


/ 


275 C374 -292 37171 


309 0170 


325 5682 


342 0201 


858 3679 


374 6066 •390 7311' 60 


9170 64991 


2936 


8432 


2935 


6395 


8763 9989 59 


276 1965 


9280 


5702 


326 1182 


5668 


9110 


3751459.3912666 58 


47C1 


293 2061 


8468 


3932 


8400 


359 1825 


4156 5343; 67 


755C 


4842 


310 1-234 


6681 


343 1133 


4540 


6852 8019' 56 


277 0352 


7623 


3999 


9430 


3805 


7254 


9547 •392 0695! 55 


3147 


294 0403 


6764 


327 2179 


6597 


9968 


376 2243, 3371] 54 


5941 


3183 


95«9 


4928 


9329 


360 2682 


4938 


6047 53 


873e 


5963 


311 2294 


7676 


3442060 


5395 


7632 


8722! 62 


278 1530 


8743 


5058 


328 0424 


4791 


8108 


377 0327 


3931397' 61 


4324 


295 1522 


0-, 0*^^22 


3172 


7521 


•361 0821 


3021 


4071 50 


7118 


4302 


312 058(3 


5919 


345 0252 


3534 


5714 


6745 


49 


9911 


7081 


3349 


8666 


2982 


62461 


8408! 


9419 


48 


279 2704 


9859 


6112 


329 1413 


5712 


8958 


378 1101 


394 2093 


47 


5497 


296 2638 


8875 


4160 


8441 


•362 1669 


3794 


4766 


46 


8290 


5416 


•313 1638 


6906 


346 1171 


4380 


6486 


7439 


45 


2801083 


8194 


4400 


9653 


3900 


7091 


9178 


395 0111. 


44 


3875 


297 0971 


7163 


330 2398 


6628 


9802 


•379 1870 


2783 


43 


6667 


3749 


9925 


5144 


9357 


•363 2512 


4562 


5455 


42 


9459 


6526 


•314 2686 


7889 


347 2085 


5222 


7253 


8127 


41 


281 2251 


9303 


5448 


331 0634 


4812 


7932 


9944 


•396 0798 


40 


5042 


•298 2079 


8209 


3379 


7540 


■364 0641 


380 2634 


3468 


39 


7833 


4856 


•315 0969 


6123 


•348 0267 


3351 


5324 


6139 


38 


282 0624 


7632 


3730 


8867 


2994 


6059 


8014 


8809 


37 


3415 


•299 0408 


6490 


•3321611 


5720 


8768 


•381 0704 


•397 1479 


36 


6205 


3184 


9250 


4355 


8447 


•365 1476 


3393 


4148 


36 


8995 


5959 


•316 2010 


7098 


•349 1173 


4184 


6082 


6818 


34 


2831785 


8734 


4770 


9841 


3898 


6891 


8770 


9486 


33 


4575 


•3001509 


7529 


•333 2584 


6624 


9599 


•3821459 


■398 2155 


32 


7364 


4284 


•317 0288 


5326 


9349 


•366 2306 


4147 


4823 


31 


2840153 


7058 


3047 


8069 


•350 2074 5012 


6834 


7491 


30 


2942 


9832 


5805 


•3340810 


4798 


7719 


9522 


.399 0158 


29 


5731 


•301 2606 


8563 


3552 


7523 


■367 0425 


•383 2209 


2825 


28 


8520 


5380 


•318 1321 


6293 


■351 0246 


3130 


4895 


5492 


27 


285 1308 


8153 


4079 


9034 


2970 


5836 


7582 


8158 


26 


4096 


•302 0926 


6836 


•3351775 


5693 


8541 


•384 0268 


•400 0825 


25 


6884 


3699 


9593 


4516 


8416 


■368 1246 


2953 


3490 


24 


9671 


6471 


•319 2350 


7256 


•3521139 


3950 


5639 


6IS6 


23 


•2862458 


9244 


5106 


9996 


3862 


6654 


8324 


8821 


22 


524e 


■303 2016 


7863 


•336 2735 


6584 


9358 


•385 1008 


•401 1486 


21 


8032 


4788 


•320 0619 


5475 


9306 


■369 2061 


3693 


4160 


20 


•287 0819 


7559 


3374 


8214 


•353 2027 


4765' 6377 


6814 


19 


3605 


•304 0331 


6130 


•337 0953 


4748 


7468 


9060, 9478 18 


6391 


3102 


8885 


3691 


7469 


•370 0170 


•386 1744 


•402 2141 


17 


9177 


5872 


•321 1640 


6429 


•3540190 


2872 


4427 


*" 4804 


16 


•2881963 


8643 


4395 


9167 


2910 


5574 


7110 


7467 


16 


4748 


•305 1413 


7149 


•338 1905 


5630 


8276 


9792 


•403 0129 


14 


7533 


4183 


9903 


4642 


8350 


•371 0977 


•387 2474 


*" 2791 


13 


•289 0318 


6953 


•3222657 


7379 


•355 1070 


3678 


6156 


6453 


12 


3103 


9723 


5411 


•339 0116 


3789 


6379 


7837 


8114 


11 


5887 


•306 2492 


8164 


2852 


6508 


9079 


•388 0518 


•404 0775 


10 


8671 


5261 


•3230917 


5589 


9226 


•3721780 


3199 


3436 


9 


•2901455 


8030 


3670 


8325 


•3561944 


4479 


5880 


6096 


8 


4239 


•307 0798 


6422 


•340 1060 


4662 


7179 


8560 


8756 


7 


7022 


3566 


9174 


3796 


7380 


9878 


•389 1240 


•405 1416 


6 


9805 


6334 


•3241926 


6531 


•357 0097 


•373 2577 


3919 


4075 


5 


•291 2588 


9102 


4678 


9265 


2814 


5275 


6598 


6734 


4 


5371 


■308 1869 


7429 


•341 2000 


5531 


7973 


9277 


9393 


3 


815S 


463e 


•325 0180 


4734 


8248 


•374 0671 


•390 1955 


•406 2051 


2 


•292 093E 


740S 


2931 


7468 


•358 0964 


3369 


4633 


4709 


1 


3717 


•309 017C 


5682 


•342 0201 


367S 


6066 


7311 


7366 





73° 


72° 


71° 


70° 
' NAT. < 


69° 
30SINE. 


68° 


1 67° 


66° 


/ 



NATURAL SINES. 



247 



24° 

•406 7366 
•407 0024 

2881 
5337 
7993 

■408 0649 
3305 
5960 
8615 

■409 1269 
3923 
65 
■ 9230 

•410 1883 
4536 
7189 
9841 

■411 2492 
5144 
7795 

•412 0445 
3096 
5745 
8395 

•413 1044 
3693 
6342 
8990 

•414 1638 
4285 
6932 
9579 

•415 2226 
48' 
7517 

•416 0163 
2803 
5453 
8097 

•417 0741 
3385 
6028 
8671 

•418 1313 
3956 
659 
9233 

•419 1880 
4521 
7161 
9801 

•420 2441 
5080 
7719 

•421 0358 
2996 
5634 
8272 

•422 0909 

3546 

6183 

65 



25° I 

422 6183 
8819 

423 1455 
4090 
6725 
9360 

424 1994 
4628 
7262 
9895 

425 2528 
5161 
7793 

426 0425 
3056 
5687 
8318 

427 0949 
3579 
6208 
8838 

428 1467 
4095 
6723 
9351 

429 1979 
46061 
7233 
9859 

430 2485 
5111 
7736 

431 0361 
2986 
5610 
8234 

432 0857 
3481 
6103 
8726 

433 1348 
3970 
6591 
9212 

434 183: 
4453 
7072 
9692 

435 2311 
4930 
7548 

430 0166 
2784 
6401 
8018 

437 0634 
3251 
5866 
8482 

•438 1097 
3711 
64° 



26° 


27° 


28° 1 


•438 3711 


•453 9905 ^469 4716 • 


6326 


•454 2497 7284 • 


8940 


50881 9852 


•4391553 


7679 -470 2419 


4166 


•455 0269 


4986 


6779 


2859 


7553 • 


9392 


5449 


•4710119 


440 2004 


8038 


2685 


4615 


•456 0627 


5250 


7227 


3216 


7815 • 


9838 


5804 


•472 0380 


441 2448 


8392 


2944 


5059 


•457 0979 


5508 


7668 


356G 


8071 • 


442 0278 


6153 


•473 0634 


2887 


8739 


3197 


5496 


•458 1325 


5759 


8104 


3910 


8321 • 


•443 0712 


6496 


•474 0882 


3319 


9080 


3443 


5927 


•459 1665 


6004 


8534 


4248 


8564 


•444 1140 


6832 


•475 1124 


3746 


9415 


3683 


6352 


•460 1998 


6242 


8957 


4580 


8801 


•445 1562 


7162 


■4761359 


4167 


9744 


3917 


6771 


■461 2325 


6474 


9375 


4906 


9031 


•4461978 


7486 


•477 1588 


4581 


•462 0066 


4144 


7184 


2646 


6700 


9786 


5225 


9255 


•447 2388 


7804 


•478 1810 


4990 


•463 0382 


4364 


7591 


2960 


6919 


•448 0192 


5538 


9472 


2792 


8115 


•479 2026 


5392 


■464 0692 


4579 


7992 


3269 


7131 


•449 0591 


5845 


9683 


3190 


8420 


•480 2235 


5789 


■465 0996 


4786 


8387 


3571 


7337 


•450 0984 


6145 


9888 


3582 


8719 


•481 2438 


6179 


•466 1293 


4987 


8775 


3866 


7537 


•4511372 


6439 


•482 0086 


3967 


9012 


2634 


6563 


•467 1584 


5182 


9158 


4156 


7730 


•4521753 


6727 


•483 0277 


4347 


9298 


2824 


C941 


■468 1869 


5370 


9535 


4439 


7916 


•453 2128 


7009 


•484 0462 


4721 


9578 


3007 


7313 


•469 2147 


5552 


9905 


4716 


8096 


63° 


62° 


61° 



29° ! 

484 8096 ■ 

485 06401 
3184J 
57-271 
8-2701' 

•486 08121 
33541 
5895! 
8436 

•487 0977 
3517 
6057 
8597 

•488 1136 
3674 
6-212 
8750 

•489 1288 
3825 
6361 
8897 

•490 1433 
3968 
6503 
9038 

•491 1572 
4105 
6638 
9171 

•492 1704 
4236 
0767 
9298 

•493 18-29 
4359 
6889 
9419 

•494 1948 
4470 
7005 
95321 

•495 2060 
4587 
7113 
9639 

•496 2165 
4690 
7215 
9740 

•497 2264 
4787 
7310 
9833 

■498 2355 
4877 
7399 
9920 

•499 2441 
4961 
7481 

■500 0000 
60° 



30° 


31° 


500 0000 


•515 0381 


2519 


2874 


5037 


5367 


7556 


7859 


•501 0073 


•516 0351 


2591 


2842 


5107 


5333 


7624 


7824 


•502 0140 


•517 03U 


2655 


2804 


5170 


5293 


7685 


7782 


503 0199 


■518 0-270 


2713 


2758 


5227 


5246 


7740 


7733 


■504 0252 


519 0219 


•2765 


2705 


5-276 


5191 


7788 


7676 


•505 0298 


■520 0161 


2809 


2646 


5319 


5130 


78-28 


7613 


■506 0338 


521 0096 


2846 


2579 


5355 


5061 


7863 


7543 


■507 0370 


•522 0024 


2877 


2505 


5384 


4986 


7890 


7466 


•508 0396 


9945 


2901 


•523 2424 


5406 


4903 


7910 


7381 


■509 0414 


9859 


2918 


■524 2336 


5421 


4813 


79-24 


7290 


•510 042C 


9766 


292S 


■525 2-241 


54-20 


4717 


793i- 


7191 


•511 0431 


9665 


2331 


•526 2139 


5431 


4613 


7930 


7085 


■512 04-29 


9558 


2927 


•527 ^2030 


5425 


4502 


7923 


697.:' 


■513 04-20 


9443 


2916 


•528 1914 


5413 


4383 


7908 


685S 


•514 0404 


9322 


2899 


•529 1790 


5393 


4258 


7887 


6726 


•515 0381 


9193 


59° 


58° 



NAT. COSINE. 



248 



NATURAL SINES. 



32° 


33° 


34° 


35° 


36° 


37° 


38° 


39° 1 


f 


629 9193 


544 6390 


559 1929 


573 5764 


587 7853 


601 8150 


615 6615 ^629 3204 


60 


530 1659 


8830 


4340 


8147 


588 0206 


e02.0473 


8907 i 5464 


59 ^ 


4125 


545 1269 


6751 


574 0529 


2558 


2795 


GIG 1198: 7724 


58 i 


G591 37071 


ylGi: 


2911 


4910 


5117 


3489 


9983 


57 ; 


9057 


0145 


560 157:^ 


5292 


7262 


7439 


5780 


•630 2242 


56 ■■ 


5311521 


8583 


39S1 


1 <j7^ 


9G13 


97 to 


8069 


4500 


55 • 


39SG 


54G 1020 


6390 


575 0053 


589 1964 


603 2080 


617 0359 


6758 


64 


6450 


345G 


879S 


2432 


4314 


4400 


2648 


9015 


63 


8913 


5392 


5G1 1206 


4811 


6663 


C71? 


4936 


•631 1272 


52 


532 137C 


8328 


3G14 


7190 


9012 


9038 


7224 


3528 


51 


3S39 


547 07G3 


0021 


9568 


-.590 1361 


604 135G 


9511 


5784 


60 


6301 


3198 


S4-2S 


576.19-U. 


3709 


3C74 


618 1798 


8039 


49 


8763 


5G32 


5G2 0G3i 


4323 


6057 


5991 


4084 


•632 0293 


48 


5331224 


8060 


3239 


C70G 


8404 


8308 


6370 


2547 


47 


3685 


548 0499 


5G4o 


9070 


•591 0750 


605 0624 


8655 


4800 


46 


6145 


2932 


8049 


•577 1452 


3096 


2940 


■619 0939 


7053 


45 


8605 


53G5 


•563 0453 


382, 


5442 


5255 


3224 


930t 


44 


5341065 


7797 


2S57 


6202 


7787 


7570 


5507 


•633 1557 


43 


352.3 


•549 02^28 


5260 


8576 


•592 0132 


9884 


7790 


3809 


42 


5982 


2659 


7G63 


•578 0950 


2476 


•COG 2198 


•620 0073 


6059 


41 


8440 


5090 


•564 0066 


3323 


4819 


4511 


. 2355 


8310 


40 


535 0898 


7520 


2467 


5696 


7163 


6824 


4636 


•634 0559 


39 


3355 


9950 


4869 


8069 


9505 


9136 


6917 


2808 


38 


5812 


•550 2379 


7270 


•579 0440 


•593 1847 


•607 1447 


9198 


5057 


37 


8268 


4807 


9670 


2812 


4189 


3758 


•621 1478 


7305 


36 


536 0724 


7236 


•565 2070 


5183 


6530 


6069 


3757 


9553 


35 


3179 


9663 


4469 


7553 


8871 


8.379 


6036 


•6351800 


34 


5fi34 


•551 2091 


6868 


9923 


•594 1211 


•608 0689 


8314 


4046 


33 


8089 


4518 


92G7 


•580 2292 


3550 


2998 


•6220592 


6292 


32 


537 0543 


6944 


•566.1665 


4661 


5889 


5306 


2870 


8537 


31 


2990 


9370 


4062 


7030 


8228 


7614 


5146 


•G360782 


30 


5449 


•5521795 


6459 


9397 


■595 0566 


9922 


74-23 


3026 


29 


7902 


4220 


8856 


■5811765 


2904 


•e09 2229 


9G98 


5270 


28 


•5380354 


6645 


567 1252 


4132 


5241 4535 


■6-23 1974 


7513 


27 


28or 


90G9 


3648 


6498 


7577 6841 


4248 


9750 


26 


5257 


•553 1492 


6043 


8864 


9913 9147 


6522 


•637 1998 


25 


7708 


3915 


8437 


•5821230 


-596 2249 •610 1452 


8796 


4240 


24 


•5390158 


G338 


•5G8 0832 


3595 


4584 3766 


•6-241069 


6481 


23 


2608 


8760 


3225 


5959 


6918 1 6060 


3342 


8721 


22 


5058 


•5541182 


0619 


8323 


9252 


6363 


5614 


•638 0961 


21 


7507 


3603 


8011 


•583 0687 


•597 1586 


•611 0666 


7885 


3201 


20 


9955 


6024 


•569 0403 


3050 


3919 


2969 


•625 015€ 


5440 


19 


•540 2403 


8444 


2795 


5412 


6251 


5270 


2427 


7678 


18 


4851 


•555 0804 


5187 


7774 


8583 


7572 


469C 


9916 


17 


7298 


3283 


7577 


•58101361-598 0915 


9873 


6966 


•639 2153 


16 


9745 


5702 


9968 


2497 3246 


•612 2173 


9235 


4390 


15 


•541 2191 


8121 


•570 2357 


4857 5577 


4473 


•6261503 


6626 


14 


4637 


•556 0539 


4747 


7217 7906 


6772 


3771 


8862 


13 


7082 


2956 


7136 


957 7 1-599 0236 


9071 


6038 


•640 1097 


12 


9527 


5373 


9524 


•585 19S6| 2565 


•613 1369 


8305 


3332 


11 


•542 1971 


7790 


•571 1912 


4294' 4893 


3C6f 


•G27 0571 


556r 


10 


4415 


•557 0206 


4299 


6652 72-21 


5964 


2837 


7799 


9 


6859 


2621 


668e 


9010, 9549 


8260 


5102 


•641 0032 


8 


9302 


5036 


9073 


•580 1367, -600 1876 


•614 0556 


7366 


2264 


7 


•543 1744 


7451 


•572U.59 


3724I 4^202 


2852 


9631 


4496 


6 


4187 


9865 


3«44 


6080 6528 


5147 


•628 1894 


6728 


5 


662S 


•558 2279 


622:1 


8435! 8854 


7442 


41.57 


8958 


4 


9069 


4G92 


8614 


•587 0790 •eOl 1179 


9736 


64-20 


•&i21189 


3 


•544 1510 


7105 


•5730098 


3145' 3503 


•615 2029 


8682 


3418 


2 


3951 


9517 


3381 


5499, 5827 


4322 


•('29 0943 


5647 


1 


6390 


•559 1929 


5764 


7853, 8150 


6615 


3204 


7876 





57° 


56° 


55° 


54° 1 53° 


52° 


51° 


50° 


/ 








NAT. ( 


liOSINE. 











KATCRAL SIXES. 



240 



40° 

642 7876 

t)43 0104 
2332 
4559 
6785 
9011 

644 1236 
3461 
5685 
7909 

645 0132 
2355 
4577 
6798 
9019 

■646 1240 
3460 
5679 
7898 

•647 one 

2334 
4551 
6767 
8984 

■648 1199 
^14 
5628 
7842 

•649 0056 
2268 



41° I 42° 

656 05901 -669 1306 
3468 



4480 
6692 
8903 

650 1114 
3324 
5533 
7742 
9951 

6512158 
4366 
6572 
8778 

•652 0984 
3189 
5394 
7598 
9801 

■653 2004 
4206 
6408 
8G09 

•654 0810 
3010 
5209 
740S 
9607 

•655 1804 
400' 
6198 
8395 

•656 0590 
49 



4980 
7174 
9367 

657 1560 
3752 
5944 
8135 

658 0326 
2516 
4706 
6895 
9083 

C'o9 1-271 
3458 
5645 
7831 

660 0017 
2202 
4386 
6570 
8754 

661 0936 
3119 
530C 
7482 
9665 

662 184-j 
4022 
6200 
8379 

•C03 0, 
2734 
4910 
7087 
9262 

•664143' 
3612 



5628 
7789 
994S 

670 2108 
4266 
6424 
8582 

671 0739 



43° 

•681 9984 
•682 2111 
4237 
6363 
8489 
■683 0613 
2738 
4861 
69S4I 
910' 



44° I 45° 

■694 6584! •707 1068 

86761 3124 

■695 0767 5180 

2858 7236 

4949 9291 



2895 


•684 12-29 


5051 


3350 


7206 


5471 


9361 


7591 


672 1515 


9711 


3668 


■685 1830 


5821 


3948 


7973 


6066 


673 0125 


8184 


2276 


•6860300 



7039 
9128 
■C9n^J17 
!.05 
5392 



708 1345 
3398 
5451 
7504 
9556 



5785 
7959 

665 0131 
2304 
4475 
6646 
8817 

■666 0987 
31 5r 
5325 
7493 
9C01 

•C67 18-28 
3994 
6160 
8326 

•608 0490 
2655 
4818 
6981 
9144 
i69 1306 

48' 



442' 

057' 
■872^ 

074 0876 
3024 
5172 
7319 
9466 

6751612 
375' 
5902 
8046 

•676 0190 
2333 
447G 
6618 
8760 

677 0901 
3041 
5181 
7320 
9459 

678 1597 
3734 
5871 
8007 

•679 0143 
2'.-7S 

4H3 

■ 6547 
8681 

■680 0813 
2946 
5078 
7209 
9339 

•681 1469 
3599 
5728 
7856 



2416 
4532 
6647 
8761 



•rs' 



47' 



08 

2988 

5101 

7213 

9325 

688 1435 
3546 
5655 
77C5 
9873 

089 1981 
4089 
6196 
830: 

•690 0407 
2512J 
461 
67-21 
8824 

•691 09-2' 
30-29 
5131 
7232 
933: 

692 1432 
3531 
5C30 
','728 
9825 
693 19-22 
4018 
6114 
8-209 

•694 0304 
2398 
4491 
6584 

46' 



479 -709 160: 
95651 365: 

697 1651 570' 
3736 775' 
58211 9806 
7905! -710 1854 
99881 3901 

698 2071 5948 
4153 7995 
6234 ^711 0041 



8315 

099 0396 

2476 

4555 



2086 
4130 
6174 

8218 
C633! •712 0260 
87111 2303 
700 0789 4344 
2866 6385 
4942; 8426 
7018 -713 0465 



9093! 

701 1167 
3^24l! 
5314' 
7387! 
9459 

7021531 
3601 
5672 
7741 
9811 

7031879 
3947 
6014 
8081 

■704 0147 
2213 
4278 
6342 
8406 

■705 
2532 
45941 
6655' 
8716' 

•700 07761 
2835 
4894 
6953' 
90111 

■707 1068 

45° 



2504 
4543 
6581 
8618 

714 0655 
2691 
4727 
6762 
8796 

715 0830 
2863 
4895 



716 
3019 
5049 
7078 
9106 

•717^1134 
3161 
518 
7213 
9238 

•718 1263 

328' 

. 5310 

7333 

9355 

•719 1377 
33£ 
440 



46° 

719 3398 
5418 
7438 
9457 

7201476 
3494 
5511 
75-28 
9544 

7211559 
3574 
5589 
7602 
9615 

7221C28 
364(> 
5651 
7661 
9671 

■723 1681 



7705 
9712 

7241719 
3724 
5729 
7734 
9738 

725 1741 
3744 
5746 
774 
9748 

7261748 
3748 
5747 
7745 
9743 

727 1740 



47° 

731 3537 
5521 
7503 
9486 

782 1467 
3449 
5429 
7409 
938^ 



.7331367 51 



3345 
5322 
7299 
9-275 

734 1250 
3225 
5199 
7173 
914C 

735 1118 



5061 
7032 
9002 

736 0971 
2940 
4908 
6875 
8842 

•737 0808 
2773 
4738 
6703 
8666 

•738 0629 
2592 
4553 
6515 
84' 

•739 0435 



3736 
5732 
7728 
97-22 
728 1716 
3710 
5703 
7695 



2394 
4353 
6311 
8268 
740 0225 
2181 
413 
6092 
8040 
■7291677/7410000 



3668 i 


1953 


10 


50571 


390r 


9 


7646 


5857 


8 


90351 


7808 


7 


730 1 023 


9758 


6 


3610-7421708 


5 


5597 


365F 


4 


7583 


5606 


3 


9568 


7554 


2 


•731 15531 


9502 


1 


35371 


743 1448 





43° 1 


42° 


/ 



NAT. COSINE 



50 



JVA run A L S INES 



48° I 49° 

•7-t3 1448 ' -754 709G 
3394 ' 9004 
5340 ' -755 0911 



728:) 
9229 i 

7411173! 
3115 
5058 
6999 
8941 

•745 0881 
2821 
4760 
669 J 
8636 

•746 0574 
2510 
4446 
0382 
8317 

•747 0251 
2184 
4117 
6049 
7981 
9912 

•748 1842 

3772 

5701 

7629 

. 9557 

•74£ 1484 
3411 
5337 
7202 
9187 

•7501111 
3034 I 
4957 I 
687 U i 
8800! 

•751 0721 
2641 ; 
4561 
6480 
8398 

•752 0316 
2233 
4149 
6065 
V980 
9894 

•753 1808 
3721 
5634 
7546 
9457 

•75U3G8 
3278 
5187 
7096 

41° 



281 X 
4724 
C630 
8535 

•756 0439 
2342 
424'J 
6148 
8050 
9951 

•757 1851 
3751 
5050 
7548 
9446 

■7581343 
3240 
5136 
7031 
8926 

•759 0820 
2713 
4606 
6498 
8389 

•760 02S0 
2170 
4060 
594;) 
7837 

•7011011 

3197 
53t)3 
7 268 
9152 
•762 1U36 



■7641714 
3590 
5465 
7340 
9214 

■765 1087 
2960 
4S32 
6704 
8574 

■766 0444 

40° 



50° 

766 0444 • 
2314 
4183 ; 
6051 
7918, 
9785 • 

■767 1652 I 
3517 
5382 i 
7246: 
9110, 

■768 0973 • 
2835 
4697 t 
6558! 
8418 1 

•769 0278 • 
2137 ' 
3996: 
5853: 
7710 I 
9567 ! 

•770 1423 1 • 
3278 
5132 I 
6986 i 
8840 1 

•771 0692 I ■ 
2544 i 
4395 



6246 
8095 i 
9945 1 

121794; 
3642 I 
5189 I 
7336; 
9182 I 

r31027 ■ 
2872 



2919 


4716 


4802 


6559 


6683 


8402 


8561 


•774 0244 


63 0445 


2086 


2325 


392o 


4-204 


5767 


6082 


7606 


7960 


9445 


9838 


•775 1-283 



3121 
4J57 
6791 
80-29 

■776 0464 
2298 
■ 4132 
6965 
7797 
9629 

•777 1460 

39° 



51° 


52° 


53° 


777 1460 


•788 0108 


•798 6355 


3290 


1898 


8105 


5120 


3688 


9855 


6949 


5477 


•799 1604 


8777 


7266 


3352 


•778 0604 


9054 


5100 


2431 


•789 0841 


6847 


4258 


2627 


8593 


6084 


4413 


•800 0338 


7909 


6198 


2083 


9733 


7983 


3827 


•779 1557 


9767 


5571 


3380 


•790 1550 


7314 


5202 


3333 


9050 


7024 


5115 


•801 0797 


8845 


6896 


2538 


•780 0665 


8676 


4278 


2485 


•791 0456 


6018 


4304 


2235 


7756 


6^23 


4014 


9495 


7940 


5792 


•802 1232 


9757 


7569 


2969 


•781 1574 


9345 


4705 


3390 


•792 11-21 


6440 


5205 


2896 


8175 


7019 


4671 


9909 


8833 


6445 


•8031642 


•782 0646 


8218 


3375 


2459 


9990 


5107 


4270 


•793 1762 


6838 


6082 


3533 


8569 


7892 


5304 


•8040299 


9702 


7074 


2028 


•7831511 


8843 


3756 


3320 


•794 0611 


5484 


51-27 


2379 


7211 


6935 


4146 


8938 


8741 


5913 


•805 0664 


•784 0547 


7678 


2389 


2352 


9444 


4113 


4157 


■795 1208 


5837 


t901 


2972 


7560 


7764 


4735 


9-283 


9566 


6497 


•806 1005 


•785 1368 


8259 


2726 


3169 


•796 0020 


4U6 


4970 


1780 


6106 


6770 


3540 


7885 


8569 


5-299 


9603 


•786 0367 


7058 


•807 1321 


2165 


8815 


3038 


3963 


•797 0572 


4754 


5759 


2329 


0470 


7555 


4084 


8185 


9350 


5833 


9899 


•787 114G 


7594 


•8081612 


2339 


9347 


3325 


4732 


•7981100 


5037 


6524 


•,^853 


6749 


8316 


4604 


8460 


•788 010S 


6355 


•809 0170 


38° 


37° 


36° 



54° 

809 0170 
1879 
3588 
5296 
7004 
8710 

810 041C 
21-22 
3826 
5530 
7234! 
8936 I 

811 0638 
2339 
4040 
5740 
7439 
9137 

812 08135 
2532 
4229 
5925 
7620 
9314 

813 1008 
2701 
4393 
6084 
7775 
9466 

8141155 
2844 
4532 
6220 
7906 
9593 

815 1-27 
2963 
4647 
6330 
8013 
9695 

8161376 
3056 
4736 
6416 
8094 
977: 

817 1449 
3125 
4801 
6476 
8151 
9S'24 

•SIS 1497 
3169 
4S41 

i--.v: 

818-J 

9852 

-819 1520 

35° 



NAT. COSINK. 



I,\iTURAL SINES. 



251 



60 



65° 


56° 


57° 


58° 


59° 


60° 


61° 


819 1520 


•8-29 0376 


•838 6706 


•848 0481 


•857 1673 


•866 0254 


•S74 6197 


3189 


2002 


8290 


2022 


3171 


1708 


7607 


4856 


3628 


9873 


3562 


4068 


3161 


9016 


6523 


5252 


•839 1455 


510-! 


6164 


4014 


•875 0425 


8189 


6877 


3037 


0641 


76C0 


60G6 


1832 


9854 


8500 


4618 


8179 


9155 


7517 


3239 


•820 1519 


•830 0123 


6199 


9717 


•858 0649 


89C7 


4&45 


3183 


1745 


7778 


•849 1254 


2143 


•867 0417 


6051 


4840 


3366 


9357 


•2790 


3635 


1866 


7455 


6509 


4987 


•840 0936 


4325 


5127 


3314 


8859 


8170 


6607 


2513 


5860 


6619 


4762 


■876 0263 


9832 


8220 


4090 


7394 


8109 


6209 


1665 


•821 1492 


9845 


5066 


8927 


9599 


7C55 


3067 


3152 


•831 14C3 


7241 


•850 0459 


•859 108S 


9100 


4468 


4811 


3080 


8816 


1991 


2576 


•868 05 44 


5SG8 


6469 


4696 


■8410390 


3522 


4064 


1988 


7268 


8127 


6312 


1963 


5053 


5551 


3431 


8660 


9784 


7927 


3536 


65S2 


7037 


4874 


•877 0004 


•8221440 


9541 


5108 


8111 


85'23 


6315 


1402 


3096 


•8321155 


6679 


9639 


•800 0007 


7750 


2858 


4751 


2768 


8249 


•851 1167 


1491 


9196 


4254 


6405 


4380 


9319 


2693 


2975 


•869 0630 


5049 


8659 


5991 


•842 1388 


4219 


4457 


2074 


. 7043 


9712 


7602 


2956 


5745 


5939 


3512 


8437 


•8231364 


9212 


45^24 


7269 


7420 


4949 


9830 


3015 


■833 0822 


6091 


8793 


8901 


6380 


•878 1222 


4666 


2430 


7657 


•852 0316 


•8610380 


■^821 


2613 


6316 


4038 


9222 


1839 


1859 


9256 


4004 


7965 


5640 


■843 0787 


3360 


3337 


■870 0691 


5394 


9614 


7252 


2351 


4881 


4815 


2124 


6783 


•824 1262 


8858 


3914 


6402 


6292 


3557 


8171 


2909 


•834 0463 


5477 


7921 


7768 


4989 


9559 


4556 


2068 


7039 


9440 


9243 


6420 


•879 0946 


6202 


3672 


8600 


•853 0958 


•802 0717 


7851 


2332 


7847 


5275 


•844 0161 


2475 


2191 


9281 


3717 


9491 


6877 


1720 


399 J 


3' < ; 


■■^710710 


5102 


•825 1135 


8479 


3273 


55uo 


5i;;-, 


2138 


6486 


2778 


•835 0080 


4838 


7u23 


oeos 


3566 


7869 


4420 


1680 


6395 


8538 


8079 


4993 


9251 


6062 


3279 


7952 


•8540051 


9549 


6419 


•880 0633 


7703 


4878 


9508 


1564 


■8631019 


7844 


2014 


9343 


6476 


•845 1004 


3077 


2488 


9269 


3394 


•826 0983 


8074 


2618 


4588 


3956 


•872 0693 


4774 


2622 


9670 


4172 


6099 


5423 


2116 


6152 


4260 


•836 1266 


5726 


7609 


6889 


3538 


7530 


5897 


2862 


7278 


9119 


8355 


4960 


8907 


7534 


4456 


8830 


•855 0627 


9820 


6381 


•881 0284 


9170 


6050 


•846 0381 


2135 


•8641^284 


7801 


1660 


•827 0806 


7643 


1932 


3643 


2748 


9221 


3035 


2440 


9^236 


3481 


5149 


4211 


•873 0640 


4409 


4074 


•837 08-27 


5030 


6G55 


5673 


2058 


5782 


5708 


2418 


6579 


8160 


7134 


3475 


7165 


7340 


4009 


81-26 


9664 


8595 


4891 


8527 


8972 


5598 


9673 


•856 lies 


■S05 0065 


6307 


9898 


•828 0603 


7187 


■847 r219 


2671 


1514 


77^22 


■882 1269 


•2234 


8775 


2765 


4173 


2973 


9137 


2638 


3864 


•838 03a3 


4309 


5674 


4430 


•874 0550 


4007 


5493 


1950 


5853 


7175 


6887 


1963 


5376 


7121 


3530 


7397 


8675 


7344 


3375 


C743 


8749 


5121 


8939 


•857 0174 


8799 


4786 


8110 


•829 0376 


6706 


•848 0481 


1673 


•866 0254 


6197 


9476 


34° 


33° 


82° 


31° 


30° 


29° 


28° 



NAT. COSINK. 



252 



NA TUBAL SINES. 



62° 


63° 


64° 


65° 


66° 


67° 


68° 


•882 947(3 


.891 00G5 


•898 7940 


•906 3078 


•913 5455 


•920 5049 


•927 1839 


•883 0841 


1385 


9215 


4307 


6637 


6185 


2928 


2206 


2705 


•899 0489 


5535 


7819 


7320 


4016 


3569 


4024 


1763 


6762 


9001 


8455 


5104 


4933 


5342 


3035 


7989 


•9140181 


9589 


6191 


6295 


6059 


4307 


9215 


1361 


.921 0722 


7277 


7656 


7975 


5578 


•907 0440 


2540 


1854 


8363 


9017 


9291 


6848 


1665 


3718 


2986 


9447 


•8840377 


•892 0606 


8117 


2S88 


4[:.: 


4116 


•928 0531 


1736 


1920 


9386 


4111 


ec7- 


5246 


1614 


3095 


3234 


•900 0654 


5333 


7247 


6375 


2696 


4453 


4546 


1921 


6554 


8422 


7504 


3778 


5810 


5858 


3188 


7775 


9597 


8632 


4858 


7166 


7169 


4453 


8995 


■915 0770 


9758 


5938 


8522 


8480 


5718 


•908 0214 


1943 


•922 0884 


7017 


9876 


9789 


6982 


1432 


3115 


2010 


8096 


•885 1230 


■893 1098 


8246 


2649 


4286 


3134 


9173 


2584 


2406 


9508 


3866 


5456 


4258 


•9290250 


3936 


3714 


•9010770 


5082 


6626 


6381 


1326 


5288 


5021 


2031 


6297 


7795 


6503 


2401 


6639 


6326 


3292 


7511 


8963 


7624 


3475 


7989 


7632 


4551 


8725 


•916 0130 


8745 


4549 


9339 


8936 


5810 


9938 


1297 


9865 


5C22 


•886 0688 


•894 0240 


7068 


■9091150 


2462 


•923 0984 


6694 


2036" 


. 1542 


8325 


2361 


3627 


2102 


7765 


3383 


2844 


9582 


3572 


4791 


3220 


8835 


4730 


4146 


■902 0838 


4781 


5955 


4336 


9905 


6075 


5446 


2092 


5990 


7118 


6452 


■930 0974 


7420 


6746 


3347 


7199 


8279 


6567 


2042 


8765 


8045 


4600 


8406 


9440 


7682 


3109 


•887 0108 


9344 


5853 


9613 


•917 0601 


8795 


4176 


1451 


•895 0641 


7105 


•910 0819 


1760 


9908 


5241 


2793 


1938 


8356 


2024 


2919 


•9241020 


6306 


4134 


3234 


9606 


3228 


4077 


2131 


7370 


5475 


4529 


■903 0856 


4432 


6234 


3242 


8434 


6815 


5824 


2105 


5635 


6391 


4351 


9496 


8154 


7118 


3353 


6837 


7546 


5460 


•931 0558 


9492 


8411 


4600 


8038 


8701 


6568 


1619 


•888 0830 


9703 


5847 


9238 


9855 


7676 


2679 


2166 


.896 0994 


7093 


•911 0438 


•918 1009 


8782 


3739 


3503 


2285 


8338 


1637 


2161 


9888 


4797 


4838 


3575 


9582 


2835 


3313 


•925 0993 


5855 


6172 


4864 


•9040825 


4033 


4464 


2097 


6912 


7506 


6153 


2068 


5229 


5614 


3201 


7969 


8839 


7440 


3310 


6425 


6763 


4303 


9024 


•889 0171 


8727 


4551 


7620 


7912 


5405 


•932 0079 


1503 


■897 0014 


5792 


8815 


9060 


6506 


1133 


2834 


1299 


7032 


•912 0008 


■919 0207 


7606 


2186 


4164 


2584 


8271 


1201 


1353 


8706 


3-238 


5493 


3868 


9509 


2393 


2499 


9805 


4290 


6822 


5151 


•905 0746 


3584 


3644 


•926 0902 


5340 


8149 


6433 


1983 


4775 


4788 


2000 


6390 


9476 


7715 


3219 


5965 


5931 


3096 


7439 


•890 0803 


8990 


4454 


7154 


7073 


4192 


8488 


2128 


■898 0276 


5088 


8342 


8215 


5286 


9535 


3453 


1555 


6922 


9529 


■ 9356 


6380 


•933 0582 


4777 


2834 


8154 


■913 0716 


•920 0496 


7474 


1628 


6100 


4112 


9386 


1902 


1635 


8566 


2673 


7423 


5389 


■906 0618 


3087 


2774 


9658 


3718 


8744 


6665 


1848 


4271 


3912 


•927 0748 


4761 


•891 0065 


7940 


3078 


5455 


5049 


1839 


5804 


27° 


26° 


25° 


■>\o 


23° 


22° 


21° 



NAT. COSINE. 



NATURAL SINES. 



25?, 



69° 


70° 


933 5804 


•939 6926 


6846 


7921 


7888 


8914 


8928 


9907 


9968 


•940 0899 


gai 1007 


1891 1 


2045 


2881 ; 


3082 


3871 


4119 


4860; 


5154 


5848 


6189 


6835 


7223 


7822 


8257 


8808 


9289 


9793 


•935 0321 


•941 0777 


1352 


1760 1 


2382 


2743' 


3412 


3724 


4440 


4705 


5468 


5686 


6495 


6665 


7521 


7644 


8547 


8621 


9571 


9598 


•936 0595 


•942 0575 


1618 


1550 


2641 


2525 


3662 


3498 


4683 


4471 


5703 


5444 


6722 


6415 


7740 


7386 


8758 


8355 


9774 


9324 


•937 0790 


•943 0293 


1806 


1260 


2820 


2227 


3833 


3192 


4846 


4157 


5858 


5122 


6869 


6085 


7880 


7048 


8889 


8010 


9898 


8971 


•938 0906 


9931 


1913 


•944 0890 


2920 


1849 


3925 


2807 


4930 


3764 


5934 


4720 



7940 


6630 


8942 


7584 


9943 


8537 


0943 


9489 


1942 


•945 0441 


2340 


1391 


3938 


2341 


4935 


3290 


5931 


4238 


6926 


5186 


0° 


19° 



71° 


72° 


945 5186 


•951 0565 


6132 


1464 


7078 


2361 


8023 


3258 


89(58 


4154 


9911 


5050 


94G0854 


5944 


1735 


6838 


2736 


7731 


3677 


8623 


4616 


9514 


5555 


•952 0404 


6493 


1294 


7430 


2183 


8366 


3071 


9301 


3958 


917 0236 


4844 


1170 


5730 


2103 


6615 


3035 


7499 


3966 


S3S2 


4897 


9264 


5827 


•953 0146 


6756 


1027 


7684 


1907 


8612 


2786 


9538 


3664 


•948 04tU 


4542 


1389 


5418 


2313 


6294 


3237 


7170 


4159 


8044 


5081 


8917 


6002 


9790 


6922 


•954 0662 


7842 


1533 


8760 


2403 


9678 


3273 


•949 0595 


4141 


1511 


5009 


2426 


5876 


3341 


6743 


4255 


7608 


5168 


8473 


6080 


9336 


6991 


•955 0199 


7902 


1062 


8812 


1923 


9721 


2784 


•950 0629 


36JG 


1536 


4502 


2443 


5361 


3348 


6218 


4253 


7074 


5157 


7930 


6061 


8785 


6963 


9639 


7865 
8766 


•956 0492 
1345 



73° 


74° 


•956 3048 


•961 2017 


3898 


3418 


4747 


4219 


5595 


5019 


0443 


5818 


7290 


6616 


8136 


7413 


8981 


8210 


9825 


9005 


957 0669 


9800 


1512 


•902 0594 


2354 


1387 


3195 


2180 


4035 


2972 


4875 


3762 


5714 


4552 


6552 


5342 


7389 


6130 


8225 


6917 


9060 


7704 


9895 


8490 


958 0729 


9275 


1562 


•363 0060 


2394 


0843 


3226 


1626 


4056 


2408 


4886 


3189 


5715 


3969 


6543 


4748 


7371 


5527 


8197 


6305 


9023 


7081 


9848 


7858 


•959 0672 


8633 


1496 


9407 


2318 


•964 0181 


3140 


0954 


3961 


1726 


4781 


2497 


5600 


3268 


0418 


4037 


7236 


4806 


8053 


5574 


8869 


6341 


9684 


7108 


•960 0499 


7873 


1312 


8638 


2125 


9402 


2937 


•965 0165 


3V48 


0927 



•951 0565 
18° 



2197 
3048 

17° 



4558 
5368 
6177 
6984 
7792 
8598 
9403 
•961 0208 
1012 
1815 
2617 

16° 



2449 
3209 
3968 
4726 
5484 
6240 
6996 
7751 
8505 
9258 

15° 



75° 

965 9258 

966 0011 
0762 
1513 
2263 
3012 
3761 
4508 
5265 
6001 
6746 
7490 
8234 
8977 
9718 

967 0459 
1200 
1939 
2678 
3415 
4152 
4888 
5624 
6358 
7092 
7825 
8557 
9288 

968 0018 
0748 
1476 
2204 
2931 
3658 
4383 
5108 
5832 
6555 
7277 
7998 
8719 
9438 

•9690157 
0875 
1593 
2309 
3025 
3740 
4453 
5167 
5879 
6591 
7301 
8011 
8720 
9428 

•970 0135 
0842 
1548 
2253 
2957 

14° 



NAT. COSINE. 



254 



NATURAL SINES. 



21 
22 
23 
24 

25 I 

26 I 

27 1 
28 
29 



76° 


77° 


78° 


79° 


80° 1 


81° 


82° 


/ 


•970 2957 


•974 3-^01 


•978 1476 


•981 6272 


•9848 078 


•9876 883 


•9902 681 


60 


36o0 


4355 


2080 


6826 


582 


•9877 338 


•9903 085 


59 


4363 


5008 


2684 


7380 


•9849 086 


792: 


489 


58 


5065 


5660 


3287 


7933 


589 


•9878 -245 


891 


57 


5766 


6311 


3889 


8485 


•9850 091 


697 


•9904 293 1 


56 


6466 


6962 


4490 


9037 


593 


•9879 148 


694 ' 55 


7165 


7612 


5090 


9587 


•9851 093 


599 


•0905 095 ! 54 


7863 


8261 


5689 


•982 0137 


593 


•9880 048 


494 


53 


8561 


8909 


6288 


068G 


•9852 092 


497 


893 


52 


9258 


9556 


6886 


1234 


590 


945 


•9906 290 


51 


9953 


•975 0203 


7483 


1781 


•9853 087 


•9881 392 


687 


60 


•9710649 


0849 


8079 


2327 


583 


838 


•9907 083 


49 


1343 


1494 


8674 


2873 


■9854 079 


•9882 284 


478 


48 


2036 


2138 


9268 


3417 


574 


728 


873 


47 


2729 


2781 


9862 


3961 


•9855 (36S 


•9883 172 


•9908 266 


46 


3421 


3423 


•979 0455 


4504 


501 


615 


659 


45 


4112 


4065 


1047 


5046 


•9856 053 


•9884 057 


•9909 051 


44 


4802 


4706 


1638 


5587 


544 


498 


442 


43 


5491 


6345 


2228 


6128 


•9857 035 


939 


832 


42 


6180 


59S5 


2818 


6668 


524 


•9885 378 


•9910 221 


41 


6867 


6623 


3406 


7206 


•9858 013 


817 


610 


40 


7554 


7260 


3994 


7744 


501 


•9886 255 


997 


39 


8240 


7897 


4581 


8282 


988 


692 


•9911 384 


38 


8926 


8533 


5167 


8818 


•9859 475 


•9887 128 


770 


37 


9610 


9168 


5752 


9353 


960 


564 


•9912 155 


36 


•972 0294 


9802 


3337 


9888 


•9860 445 


998 


540 


35 


0976 


•976 0435 


6921 


•983 0422 


929 


•9888 432 


923 


34 


1658 


1067 


7504 


0955 


•9861 412 


865 


•9913 306 


33 


2339 


1699 


8086 


1487 


894 


•9889 297 


688 


32 


3020 


2330 


8667 


2019 


•9862 375 


728 


•9914 069 


31 


3699 


2960 


9247 


2549 


856 


•9890 159 


4i9 


30 


4378 


3589 


9827 


3079 


•9863 336 


588 


828 


29 


5056 


4217 


•980 0405 


3608 


815 


•9891017 


•9915 206 


28 


5733 


4845 


0983 


4136 


•9864 293 


445 


584 


27 


6409 


5472 


1560 


4663 


770 


872 


961 


26 


7084 


6098 


2136 


5189 


•9865 246 


•■9892 208 


•9916 337 


25 


7759 


6723 


2712 


5715 


722 


723 


712 


24 


8432 


7347 


3286 


6239 


•9866196 


•9893 148 


•9917 086' 


23 


9105 


7970 


3860 


6763 


670 


572 


459 


22 


9777 


8593 


4433 


7286 


•9867 143 


994 


832 


21 


•973 0449 


9215 


5005 


7808 


615 


•9894 41G 


•9918 204 


20 


1119 


9830 


6576 


8330 


•9868 087 


838 


574 


19 


1789 


•977 045G 


6147 


8850 


557 


•9895 258 


944 


18 


2458 


1075 


6716 


9370 


•9869 027 


677 


•9919 314 


17 


3125 


1693 


7285 


9889 


496 


•9896 096 


682 


16 


3793 


2311 


7853 


•984 0407 


9C4 


514 


•9920 049 


15 


4458 


2928 


8420 


0924 


•9870 431 


931 


416 


14 


5124 


3544 


89SG 


1441 


897 


•9S97 347 


782 


13 


5789 


4159 


9552 


1956 


•9871 363 


762 


•9921 147 


12 


6453 


4773 


•981 0116 


2471 


827 


•9898177 


511 


11 


7116 


5386 


0680 


2985 ■•9872 291 


590 


874 


10 


7778 


6999 


1243 


3498 754 


•9899 003 


•9922 237 


9 


8439 


6611 


I 1805 


4010 


•9873 216 


415 


599 


8 


9100 


7222 


2306 


4521 


678 


826 


959 


7 


9760 


7832 


i 2927 


5032 


•9874138 


•9900 237 


•9923 319 


6 


•9740419 


8441 


! 3486 


5542 


598 


646 


679 


5 


1077 


9050 


4045 


6050 


•9875 057 


•9901 055 


•9924 037 


4 


1734 


9658 


4603 


6558 


514 


462 


394 


3 


2390 


•978 0265 


5160 


7066 


972 


869 


751 


2 


3046 


0871 


i 5716 


7572 


•9876428 


•9902 275 


•9925 107 


1 


3701 


1476 


I 6272 


8078 


883. 


681 


462 





13° 


12° 


1 11° 

Ni 


10° 

VT. C08IN 


9° 

E. 


8° 


70 


/ 



NATURAL SIKES. 



25ri 



83^ 


84° 


85° 


8G° 


87° 


88° 


89° 


/ 


■9925 402 


'9945 219 


•9961 947 


•9975 641 


•eS86 295 


9993 CC8 


■99'c-8 477 


60 


816 


523 


•9962 200 


843 


447 


9994 009 


5i7 


59 


•9926 16'J 


825 


452 


•9976 045 


598 


110 


577 


58 


521 


•9946127 


704 


245 


748 


209 


625 


57 


■ 873 


428 


954 


445 


898 


308 


€73 


56 


•9927 22i 


729 


■9963 204 


645 


•9987 046 


405 


720 


55 


573 


•9947 028 


453 


S43 


194 


502 


7 CO 


54 


922 


327 


701 


•9977 040 


340 


698 


812 


53 


•9928 271 


625 


948 


237 


486 


€93 


856 


52 


618 


921 


•9964195 


433 


631 


788 


900 


51 


965 


•9948 217 


440 


627 


775 


881 


942 


50 


•9929 310 


513 


685 


821 


919 


974 


964 


49 


656 


807 


929 


■9978 015 


•9988 061 


•9995 066 


•9999 026 


48 


999 


•9949 101 


•9965 172 


207 


203 


157 


065 


47 


•9930 342 


393 


414 


399 


344 


247 


105 


46 


685 


685 


655 


589 


484 


236 


143 


45 


•9931 026 


976 


895 


779 


623 


424 


l&l 


44 


367 


•9950 260 


•9966135 


9L8 


761 


512 


218 


43 


706 


556 


374 


•9979 15C 


£99 


599 


254 


42 


•9932 045 


SW 


612 


343 


•9989 C35 


184 


289 


41 


384 


•9951 132 


849 


530 


171 


770 


323 


40 


721 


419 


•9967 085 


716 


306 


854 


357 


39 


•9933 057 


705 


321 


900 


440 


937 


389 


38 


393 


990 


555 


•9980 084 


573 


•9996 020 


421 


37 


728 -9952 274 


789 


2l7 


706 


101 


452 


36 


•9934 062 


557 


•9968 022 


450 


837 


182 


482 


35 


395 


840 


254 


631 


968 


262 


511 


34 


727 


•9953 122 


485 


811 


•£990 068 


341 


5S9 


33 


•9935 058 


403 


715 


991 


227 


419 


567 


32 


389 


683 


945 


•9981 170 


355 


497 


593 


31 


719 


962 


•9969173 


348 


482 


573 


619 


30 


•9936 047 


•9954240 


401 


525 


6(..9 


649 


644 


29 


375 


517 


628 


701 


734 


724 


6C8 


28 


703 


794 


854 


877 


859 


798 


692 


27 


•9937 029 


•9955 070 


•9970 080 


•9982 052 


983 


871 


714 


2C 


355 


345 


304 


225 


•9991 106 


943 


736 


25 


679 


620 


528 


398 


228 


•9997 015 


756 


24 


•9938 003 


893 


750 


570 


350 


086 


776 


2\, 


326 


■9956165 


972 


742 


470 


156 


795 


2-2 


648 


437 


•9971193 


912 


590 


224 


813 


21 


969 


708 


413 


•9983 082 


709 


292 


831 


2C 


•9939 290 


978 


CSS 


250 


827 


360 


847 


IS 


610 


1 ^9957 247 


851 


US 


944 


426 


863 


18 


928 


; 515 


•9972 0G9 


i85 


•9992 0(0 


492 


878 


17 


■9940 246 


783 


286 


751 


176 


656 


892 


le 


5C5 


•9958 049 


502 


917 


290 


620 


905 


\l 


880 


315 


717 


•9984 OSl 


404 


683 


917 


14 


•9941 1 95 


580 


931 


245 


517 


745 


928 


ic 


510 


844 


•9973145 


408 


629 


807 


939 


li 


823 


•9959 107 


357 


570 


740 


867 


949 


n 


•9942 136 


370 


5C9 


731 


851 


927 


958 


i( 


448 


631 


780 


891 


960 


986 


966 


< 


7C0 


892 


990 


■9985 050 


•9993 069 


•9998 044 


973 


J 


•9943 070 


•9960 152 


•9974199 


209 


177 


101 


979 


' 


379 


411 


408 


367 


284 


157 


985 




6f8 


C69 


615 


524 


390 


213 


9S9 




993 


926 


822 


680 


495 


267 


993 


' 


•9944 303 


•9961183 


•9975 028 


835 


600 


321 


996 




009 


438 


233 


989 


704 


374 


998 




914 


693 


437 


•9986143 


806 


426 


1-0000 000 




•9945 219 


947 


641 


295 


908 


477 


000 




6° 


I 5° 


1 4° 


3° 

AT. COSI 


2° 
NB. 


1° 


0« 


' 



256 



NATURAL TANGENTS. 



0° 


1° 


•000 0000 


•017 4551- 


2909 


7460- 


5818 


•018 0370 


8727 


3280 


•001 1636 


6190 


4544 


9100 


7453 


•019 2010 


•002 0362 


4920 


3271 


7830 


6180 


•020 0740 


9089 


3650 


•003 1998 


6560 


4907 


9470 


781(i 


•021 2380 


•004 0725 


5291 


3634 


8201 


6542 


•0221111 


9451 


4021 


•005 2360 


6932 


5260 


9842 


8178 


023. 2753 


•006 1087 


5663 


399( 


8574 


6905 


•02H484I 


9814 


4395 


•007 2723 


7305 


5632 


•025 0216 


8541 


3127 


•008 1450 


6038 


4360 


8948 


7260 


•026 1859 


•009 017? 


4770 


3087 


7681 


5996 


•027 05921 


8905 


3503 


•010 1814 


6414 


4724 


9325 


7C3G 


•028 2236 


•Oil 0542 


5148 


3451 


8059 


63C1 


•029 0970 


9270 


3882 


•012 2179 


6793 


5088 


9705 


799? 


•030 2616 


•013 0907 


5528 


3817 


8439 


072r 


■031 1351 


9636 


4263 


•014 2545 


7174 


5454 


•032 0086 


8364 


2998; 


•015 127S 


5910| 


4183 


88221 


7093 


•033 17341 


•016 0002 


4646! 


2912 


7558 


582] 


•034 0471! 


8731 


3383' 


•017 1641 


6295 '• 


4551 


9208 j 


89° 


88° 1 



2° I 

034 9208-05 

035 21-20 
5033' 
7945' 

030 0858: 

37711 

6683 

95961 
037 2500, 

5422 



8335 

038 1-248| 
4161' 
7074 
9988, 

039 2901' 
5814 
8728 

040 lG4i; 
4555 
7409' 

041 0383 
3296 
6-210 
9124 

042 2038 
4952' 
7866: 

043 07811 
36951 



4078 
6995 
9912 

•053 2829 
5746 
8663 

•054 1581 
4498 
7416 

■055 0333 
3251 
6169 
908' 

•056 2005 
4923 
7841 

-057 0759 
3678 
0596 
9515 

•058 2431 
5352 
8271 

-059 1190 
4109 
7029 
9948 

•060 2867 
5787 



6609 


061 1626 


9524: 


4546 


044 2438! 


7466 


5353 


062 0386 


8268: 


3306 


045 1183; 


6226 


4097! 


9147 


7012-063 2067 


9927 1 


4988 


046 2842 


7908 



5757: 

8673 

■047 1588 
4503: 
7419 

■048 0334' 
3250; 
6166' 
9082 

0491997' 
4913 
7829 

■050 0740, 
3662 
6578 
9495 

051 2411 
5328, 
8-244 

0521161 

4078 

87° I 



-064 0829 
3750 
0671 
9592 

-065 2513 
5435 
8356 

-0661-278 
4199 
7121 

-067 0043 
2965 
5887 
8809 

-068 1732 
4654 
7577 

-069 0499 
3422 
6345; 
9268 



069 9268 

070 2191 
5115 

8038 

■071 0:)61 

3885 

6S09 

072 2657 
5581 
8505 

■073 1430 
4354 
7279 

■074 0203 
3128 
6053 
8979 

■075 1904 
4829 
7755 

■076 0680 
3606 



087 4887 
7818 

088 0749 
31)81 
6612 
9544 

089 '2476 
5408 
8341 

090 1273 
4200 
7138 

091 0071 
3004 
5938 
8871 

092 1804 
4738 
7672 

093 0606 
3540 
6474 
9409 

6532 -094 2344 
9458 5278 

077 2384 8213 
5311 -095 1148 
8237 4084 

078 1164 7019 
40901 9955 



7017; 
9944 

-079 2871! 
5798! 
87261 

-080 1653 1 
4581 
7509 

•081 0437 
3365 
6293 
9221 

•082 2150 
5078 
8007 

•083 0936 
3865 
6794 
9723 

•084 2653 
5583: 
8512| 

■085 1442 
4372 
73021 

086 0233' 
31631 
6094 
9025' 

087 1956i 
4887 

85° I 



•096 2890 
5826 
8763 

-097 1 
4635 
7572 

-098 0509 
3446 
6383 
9320 

-099 2257 
5194 
8133 

•100 1071 
4009 
694 
9886 

•101 2824 
5763 
8702 

•1021041 
4580 
75-20 

■103 0460 
3399 
6340 
9280 

104 2220 
5161 
8101 

105 1042 
84° 



6° 


70 


105 1042 


•122 7846 


3983 


-1-23 0798 


6925 


3752 


9866 


6705 


106 2808 


9658 


5750 


-1-24 2612 


8692 


5560 


107 1634 


8520 


4576 


125 1474 


7519 


4429 


108 0462 


7384 


3405 


-120 0339 


6348 


3294 


9-291 


6249 


109 2234 


9205 


5178 


-127 2161 


8122 


5117 


-110 1060 


8073 


4010 


-128 1030 


6955 


3986 


9899 


6943 


-111 2844 


9900 


5789 


■129 2858 


8734 


5815 


-1121680 


8773 


4625 


•130 1731 


7571 


4690 


-113 0517 


7648 


3463 


•131 0607 


6410 


3566 


9356 


6525 


-114 2303 


9484 


5250 


'132 2444 


8197 


5404, 


-115 1144 


8364 


4092 


-133 1324 


7039 


4285 


9987 


7246 


-116 2936 


-134 0207 


5884 


3168 


8832 


6129 


■117 1781 


9091 


4730 


-135 2053 


7679 


5015 


-118 0628 


797r 


3578 


-136 0940 


6528 


3903 


9478 


G80C 


119 2428 


9830 


5378 


•137 2T93 


8329 
120 1279 


5757 

8721 



4-230 
7182 

1210133 
3085 
6036 
89S8 

12219411 
4893 
7846; 

83° 



•138 1685 
4650 
7615 

139 0580 
3545 
C510 
9476 

140 2442 
5408 

82*^ 



NAT. COZAN. 









NA run A L TANGENTS. 






257 


' 1 80 


9° 1 10° 


11° 


12° 13° 1 U° 1 15° 1 


/ 


-140 5408 


158 38441.176 3270 


194 3803 


•212 556«i -23 1868-2 


249 3280-267 9492 


60 


1 1 S:i75 


6826! 6269 


6822 


8606 -231 1746 


6370-268 2610 


59 


2 -141 134- 


98JJJ 9209 


9841 


-213 1647 4811 


9460] 57 28 


58 


3 1 4oJ< 


153 2791-177 2269 


195 2861 


4688 7876 


250 2551! 8847 


57 


4 1 727^ 


5774 5270 


5881 


7730-232 0341 


5642, -269 1967 


56 


5 -U'iO-itJ 


8757 8270 


8901 


214 0772 4007 


87341 5087 


55 


6 


3211 


11.01740-1781271 


196 1922 


3814- 7073 


251 1826| 8-207 


54 


7 


617'.. 


47241 4-273 


4*4:3 


6857-233 01401 


4919-270 1328 


53 


8 


9147 


77081 7274 


7964 


99001 


3207 


8012 4449 


52 


9 


U3 2115 


161 0692'-179 0276 


197 0981: 


215 2944 


6274 


25211061 . 7571 


51 


10 


50S4 


3677 


3-279 


400? 


5988 


9342 


4200 -271 0694 


50 


11 


8053 


6662 


0281 


7031 


9032 


234 2410 


7^294 ' 3817 


49 


12 •14H0J-: 


96 i7 


92.^ 


•198 0053 


■210 2077 


5479 


253 0383 6940 


48 


13 1 3a J 1 


1G2 2632 -ISO -2287 


3076 


5122 


8548 


3484 -272 0064 


47 


14 1 6901 


5ol8 5291 


6100 


8167 


235 1617 


(;-5S0 3188 


46 


15 1 9J31 


8603 8295 


9124 


-217 1213 


4687 


9676, 6313 


45 


Id .•145 2;>Jl 


-ir;3 15901-181 1-299 


•199 2148 


4259 


7758 


254 2773, 9438 


44 


17 


5872 


4576 


4303 


5172 


7306 


236 08-29 


5870-273 2564 


43 


18 


88 ii 


7563 


7308 


8197 


-218 0353 


3900 


89G8 5690 


42 


19 


14e 1813 


•164 0550! 


182 0313 


-200 1222 


3400 6971 


-255 -2066; 8817 


41 


20 


4784 


3537 


3319 


4248 


6448 


.237 0044 


5165-2741945 


40 


21 


7756 


6525 


63-24 


7-274 


9496 


3116 


8264 5072 


39 


22 


•147 0727 


9513 


9330 


-201 0300 


■219 2544 


6189 


•253 1363 8201 


38 


23 


36J9 


-105 2501 


183 2337 


3327 


5593 


9'262 


4163 -275 1330 


37 


24 


0G72 


5489 


5343 


6354 


864.3 


-238 2336 


7564 4459 


36 


25 


9644 


8478 


8350 


9381 


•220 1692 


5410 


•257 0664 7589 


35 


26 


•148 2617 


•106 1467 


•184 1358 


-202 2409 


4742 


84S5 


37 66 


-276 0719 


34 


27 


5590 


4456 


4365 


5437 


7793 


•2391560 


6868 


3850 


33 


is 


8563 


7446 


7373 


8465 


•2-21 0844 


4635 


9970 


6981 


32 


29 


•149 1536 


•107 0436-185 0382 


•203 1494 


3S95 


7711 


•258 3073 


■277 0113 


31 


30 


4510 


3420 3390 


45-23 


6947 


-240 0788 


6176 


3245 


3C 


31 


7484 


6417 .6399 


7552 


9999 


3864 


9280 


6378 


29 


32 


•150 0458 


9407 9409 


-204 0582 


•222 8051 


6942 


•259 2384 


9512 


28 


33 


3433 


•168 23381-186 2418 


3612 


6104 


-241 0019 


5488 


•278 2646 


27 


34 


6408 


5390 54-28 


6643 


9157 


3097 


8593 


5780 


26 


35 


9383 


8381 8439 


9674 


-223-2211 


6176 


•260 1699 


8915 


25 


36 


•151 2358 


•169 1373 ^•187 1449 


•205 2705 


5265 


9255 


4805 


-279 2050 


24 


37 


5333 


4366! 4460 


5737 


8319 


-242 2334 


7911 


5186 


23 


38 


8303 


7358 


7471 


8769 


-2241374 


5414 


•261 1018 


8322 


22 


39 


•152 1285 


•170 0351 


•188 0483 


•2061801 


4429 


8494 


4126 


-280 1459 


21 


40 


4262 


3344 


3495 


4834 


7485 


-2431575 


7234 


4597 


20 


41 


7238 


6338 


6507 


7867 


•225 0541 


4656 


•262 0342 


7735 


19 


42 


•153 0215 


9331 


9520 


•207 0900 


3597 


7737 


3451 


-281 0873 


18 


43 


3192 


•171 23-25i-189 2533 


3934 


6654 


-244 0819 


6560 


4012 


17 


44 


6170 


5320 5546 


6988 


9711 


3902 


9670 


7152 


16 


45 


9147 


8314 8559 


•208 OOOC 


•226 -2769 


6984 


•263 2780 


•282 0292 


15 


46 


154 2125 


•172 1309 -1901573 


3035 


5827 


-245 0068 


5891 


3432 


14 


47 


510C 


43041 4587 


607.: 


8885 


3151 


9002 


6573 


13 


48 


808L 


7300 


7 60-: 


9103 


•227 1944 


6236 


•264 2114 


9715 


12 


49 


•1551061 


•173 0296 


-191 0617 


•209 214£ 


5003 


9320 


5226 


•283 2857 


11 


50 


404C 


3292 


363L 


5181 


8063 


-246 2405 


8339 


5999 


10 


51 


7019 


6288 


664? 


821? 


•22s 1123 


5491 


•265 14521 9143 


9 


52 


9398 


9285 


9664 


••210 125£ 


4184 


85771 4566-2842286 


8 


53 


•156 237? 


•174 2282 


•192 268C 


429C 


7244 


-24- 1663: 7680 5430 


7 


54 


595S 


5279 


569f 


7331 


•229 030fi 


4750,-266 0794 8575 


6 


55 


C33: 


8^277 


871C 


•211 036' 


3367 


7837i 3909|-2851720 


5 


56 


■157 191C 


•175 1275 


-1931731 


340- 


6429 


-24« 09-25' 7025 4866 


4 


57 


490C 


4273 


4745 


644^ 


949-2 


4013-267 0141' 8012 


3 


58 


7881 


7272 


776( 


) 94Sf 


•230 -2555 


710-2 


3-257:-2861159 


2 


59 


•158 086C 


•176 0-271 


-194 078-1 


\ -2X2 25-2. 


5618 


•249 0191 


6374 4306 


1 


60 


384^ 


[ 3270 


3Sor 


5 656f 


8682 


328C 


9492 7454 





/ 


81° 


80° 


79° 


78° 


77° 


76° 


75° 74° 


/ 










NAT. 


COTAN. 











258 



NATURAL TANGENTS. 



/ 1 


16° 


17° 


18° 


19° 


20° 


21° 


22° 


23° 


/ 





286 7454 


305 7307 


324 9197 


344 327G 


3G3 9702 


383 8640 


404 0262 


•424 4748 


60 


1 


287 0602 


306 0488 


325 2413 


6530 


364 2997 


384 1978 


3646 


8182 


59 


2 


3761 


3670 


5630 


9785 


6291 


5317 


7031 


■425 161C 


58 


3 


6900 


6852 


8848 


345 3040 


9588 


8656 


4050417 


5051 


57 


4 


288 0050 


307 0034 


326 206t, 


6296 


365 2885 


385 1996 


3804 


8487 


56 


5 


3201 


3218 


5284 


9553 


6182 


5337 


7191 


•4261924 


55 


6 


6352 


6402 


8504 


346 2810 


M80 


8679 


406 0579 


5361 


54 


7 


9503 


9586 


327 1724 


6068 


366 2779 


386 2021 


3968 


8800 


63 


8 


289 2655 


308 2771 


4944 


9327 


6079 


5364 


7358 


•427 2239 


52 


9 


5808 


5957 


8165 


347 2586 


9379 


8708 


•407 0748 


5680 


61 


10 


8961 


9143 


328 1387 


5846 


367 2680 


387 206G 


4139 


9121 


50 


11 


290 2114 


30£ 2330 


4610 


9107 


5981 


5398 


7531 


•428 25G3 


49 


12 


5269 


5517 


7833 


348 2368 


9284 


8744 


•408 0924 


6005 


48 


13 


8423 


8705 


329 1050 


5630 


368 258-, 


388 2'J9] 


4318 


9449 


47 


14 


291 1578 


310 1893 


4281 


8893 


589( 


543^! 


7713 


•429 2894 


46 


15 


4734 


5083 


7505 


349 21 5C 


919L 


87 s; 


•409 1108 


6339 


45 


16 


7890 


8272 


330 0731 


5420 


369 2500 


■389 213C 


4504 


9785 


44 


17 


292 1047 


311 1462 


3957 


8685 


580t 


5486 


7901 


•430 3232 


43 


18 


4205 


4653 


7184 


350 1950 


9112 


8837 


■410 1299 


6680 


42 


19 


7363 


7845 


■3310411 


521G 


■370 2420 


■390 2189 


4697 


•431 0129 


41 


20 


•293 0521 


•312 1036 


3039 


8483 


5728 


5541 


8097 


3579 


40 


21 


3080 


4229 


68L& 


351 1750 


9036 


8894 


•411 1497 


7030 


39 


22 


6839 


7422 


•332 0097 


5018 


•371 234! 


■391 2247 


4898 


•432 0481 


38 


2? 


9999 


■313 0616 


3327 


8287 


565t 


5602 


8300 


3933 


37 


1'4 


•294 3160 


3810 


6557 


3521556 


8967 


8957 


•4121703 


7386 


36 


25 


6321 


7005 


9788 


4826 


•372 227 S 


•392 2313 


5106 


•433 0840 


35 


26 


9483 


•314 0200 


■333 3020 


80De 


559:; 


5070 


8510 


4295 


34 


27 


•295 2645 


3396 


6252 


.3„136S 
^^^ 4640 


8903 


9027 


•413 1915 


7751 


33 


28 


5808 


6593 


9485 


.373 2217 


■393 2386 


5321 


.434 1208 


32 


29 


«971 


9790 


•334 2719 


7912 


5532 


5745 


8728 


4665 


31 


30 


•296 2135 


•315 2988 


5953 


•354118! 


8847 


9105 


•414 2136 


8124 


30 


31 


5299 


6186 


9188 


41;:.. 


■C74 2lL.. 


■394 24C5 


5544 


•435 1583 


29 


32 


8464 


93S5 


•335 2424 


7734 


547: 


5S-:'i 


8C53 


5043 


28 


33 


•297 1630 


•316 2585 


5660 


•3551010 


87 i.-. 


91 1 9 


•415 2CG3 


8504 


27 


34 


479G 


5785 


889C 


42SL 


■375 211i 


■395 25C2 


5774 


•436 196C 


26, 


35 


7962 


8986 


•336 2134 


7502 


54:JC 


591C 


91SC 


54-29 


25 


36 


■298 1129 


•317 2187 


5372 


■356 0840 


8753 


9280 


•410 2598 


8893 


24 


37 


4297 


5389 


8610 


411S 


•37G 2073 


•3962C4C 


6012 


•437 2357 


23 


38 


7465 


8591 


•337 1850 


7397 


5394 


6on 


9426 


68-23 


22 


39 


•299 0634 


■318 1794 


509G 


■357 0fi7C 


871G 


9376. 


■417 2841 


9289 


21 


40 


3803 


4998 


8330 


39oC 


■377 2038 


■337 274C 


6257 


•438 2756 


20 


41 


6973 


8202 


•338i571 


7237 


5361 


6114 


9C73 


6224 


19 


42 


•300 0144 


•319 1407 


4S1L 


■358 OOlf 


8C85 


9483 


-418 3091 


9693 


18 


43 


3315 


4C13 


805; 


3801 


■378 2010 


•398 2853 


6509 


•439 3163 


17 


44 


6486 


7819 


•339 129'J 


70S3 


5335 


6224 


9928 


6634 


16 


45 


9658 


•320 1025 


4543 


•359 0367 


8G61 


959i 


•419 0348 


•440 0105 


15 


46 


•301 2831 


4232 


7787 


3GG1 


■3791988 


•399 23Cf 


C7C9 


3578 


14 


47 


6004 


7440 


•340 1032 


693C 


5315 


634] 


■.420 CI 90 


7051 


13 


48 


9178 


•321 0649 


427 £ 


•360 0222 


8C44 


97K 


3C1L 


■441 052C 


12 


49 


•302 2352 


385S 


7524 


350!: 


•380 1973 


•400 3089 


703C 


4001 


11 


50 


5527 


7067 


.341 0771 


f705 


5302 


6465 


•421 04CU 


7477 


10 


51 


8703 


•322 0278 


4C19 


■3C1 oos': 


sens 


9841 


3885 


4420954 


9 


52 


•3031879 


3489 


72C7 




■38119G4 


.401 3218 


7311 


4432 


8 


53 


5055 


C70C 


•342 05K 


crc( 


529C 


C59f 


•422 0738 


7910 


7 


54 


8232 


991;^ 


37 To 


9940 


8629 


9974 


4165 


•4431390 


6 


5o 


•3041410 


•323 3125 


701 r 


•362 324f 


■3821962 


•402 3354 


7594 


4871 


5 


56 


4588 


633? 


•343 026C 


6531 


529C 


6734 


•4231022 


8352 


4 


57 


7767 


9552 


3518 


9S23 


8631 


•403 0115 


4452 


•4441834 


3 


58 


•305 094r 


•324 276f 


6770 


•363 3115 


•3831967 


3496 


788^4 


5318 


2 


59 


4126 


5981 


•3440023 


640S 


5303 


6879 


•4241 31 C 


8802 


1 


60 


7307 


9197 


3276 


9702 


8640 


•404 0262 


474S 


•445 2287 





/ 


. 73° 


72° 


71« 


70° 
WAT. ( 


69° 
?07An 


68' 


07« 


§6° 


/ 



NATURAL TAA'GE2^TS. 



250 



24° I 

445 2287 1 • 
57731 
9260- 

446 27471 
6236: 
9726- 

447 3216: 
6708; 

448 0200' 
3693 j 
7187! 

449 0682 • 
4178 1 
7675 

4501173I' 
4672: 
817 1! 

451 1G72;' 
5173! 
807 6 1' 

452 2179 
50831 
9188 

453 2694 1 
62011 
97091 

454 3218' 
6728i 

455 0238' 
3750 
7263 

456 07761 



4290 
7806 

457 13221 
4839 1 
8357 i' 

458 1877 1 
53971 
8918,' 

459 2439' 
5962' 
9486 

400 3011 
6537; 

4010063: 
3591! 
7119 

462 0649 
4179 
7710 ' 

4631243 
4776 
8310 

4641845: 
5382 



465 2457; 
5996 
9536 

466 30771 

65' 



25'' I 

■466 3077 
6618 

467 0161 
3705 
7250 

•468 0796 
4342 
7890 

469 14391 
4988J 
8539, 

470 2090' 
5643, 
9196 

•471 2751 
6306 
9863: 

'472 3420 
6978 

'473 0538' 
4098' 
7659, 

•474 1222 
4785 
8319 

•475 1914' 
5481, 
9J4S 

•476 2616 
6185, 
9755' 

•477 3326' 
689d 

•478 0472' 
4046, 
7621 

■479 1197 
4774 
8352 

•480 1932J 
5512, 
9093 

•481 2675 
6258, 
9842 

•482 3427 
7014 

•483 0601 
4189 
7778 

•434 1368 
4959 
8552| 

•485 21451 
6739 
93341 

•486 2931 
6528' 

•487 0126, 
3726 
73261 

64« 



26* 

■487 7326 

488 092 

4530 

8133 

•489 173 

5343 



61661 

9775| 

4913386'^. 



•492 0610 
4224' 
7838 

•493 1454'' 
5071 1 
8689! 

•494 2308!' 
5928 
9549 

•495 3171 
6794 

•496 0418, 
4043' 
76691 

•49712971 
4925 
8554 

•498 2185 
6816 
9449 

•499 3082 
6717 

•500 0352 
3989 
7627 

•501 12661 
49061 
S547| 

■502 2189! 
5832! 
9476: 

•503 31211 
6768i 

■504 0415: 
40631 
7713 

■505 1363 
5015, 
8668' 

•506 2322 • 
5977 
9633 ■ 

507 3290 
6948 

508 0607 • 
4267 
7929 

509 1591 ■ 
5254 

63« I 



27° 


28° 


509 5254 


•531 7094 • 


8919 


•532 0826 


510 2585 


4559 • 


6252 


8293 


9919 


•533 2029 


511 3588 


5765 • 


7259 


9503 


5120930 


•534 3242 


4602 


6981 • 


8275 


•535 0723 


513 1950 


4465 ■ 


5625 


8208 


9302 


•536 1953 


514 2980 


5699 • 


6658 


9446 


515 0338 


•537 3194 • 


4019 


6943 


7702 


•538 0694 


5161385 


4145 • 


50C9 


8198 


8755 


•539 1952 


517 2441 


5707- 


6129 


9464 


9818 


•540 3221- 


518 3508 


6JS0 


7199 


•541 0740 


519 0891 


4501 • 


4584 


8263 


8278 


•542 2027 • 


520 1974 


5791 


5671 


9557 


9368 


•543 33-24 • 


521 3067 


7092 


6767 


-544 0862 


522 0468 


4C32 - 


4170 


8404 


7874 


-545 -2177 • 


523 1578 


5951 


5284 


9727 


8990 


•546 3503 • 


524 2398 


7281 


6407 


■547 1060 • 


525 0117 


4840 


3829 


8621 


7541 


•548 2404 


•5261255 


6188 


4969 


9973 


8685 


•549 3759 


•527 2402 


7547 


6120 


•550 1335 


9839 


5125 


•528 3560 


8910 


7281 


•551 2708 


•529 1004 


6502 


4727 


•552 0297 


8452 


4093 


•530 2178 


7890 


5906 


-553 1688 


9634 


5488 


•531 3364 


9288 


7094 


-554 3091 


62^ 


6P 



29° 


30° 


31° 


f 


554 3091-577 3503 


•600 8606 


60 


6894 738-i 


•601 2566 


59 


555 0698 ^578 1262 


65^27 


58 


4504 5144 


•602 0490 


57 


8:-.ll 9027 


4464 


56 


556 2119-579 2912 


8419 


55 


6929 0797 


-G03 2386 


54 


9739 -580 Ot)»4 


6364 


53 


-557 35511 457 o 


-604 03'23 


52 


7364 8402 


4294 


51 


-558 1179-5812353 


8266 


50 


4d94 02^5 


-605 2240 


49 


8811 -682 0139 


6215 


48 


•559 2629 4034 


-606 0192 


47 


6449 7930 


4170 


46 


•560 0269 •5831S2S 


8149 


45 


4091] 572b 


-607 2130 


44 


7914! 9627 


6112 


43 


•561 1738 •584 3528 


•608 0095 


42 


55641 743J, 


4080 


41 


9391h5S5 1335 


8067 


40 


•562 3219! 5241 


■609 2054 


39 


7048| 9L48 


6043 


38 


•563 0J79/5S6 305u 


■610 0034 


37 


4710 6965 


4026 


36 


S543;5S7 0870 


8019 


35 


•5612:378 4788 


■611 2014 


34 


0J13 8702 


6011 


33 


-565 0J50:-588 2616 


•612 0008 


•32 


3888 


6533 


4007 


31 


7728 


•589 0450 


8008 


30 


•566 1568 


4369 


•613 2010 


"29 


5410 


8289 


6013 


28 


9254 


•590 2211 


•6140018 


27 


-567 3J98 


6134 


4024 


26 


6944 -591 0058 


8032 


25 


-568 0791 3984 


•615 2041 


24 


4639 7910 


6052 


23 


8488-5921839 


■616 0064 


22 


•569 2339 5768 


4077 


21 


6191 9699 


8092 


20 


•570 0045! •593 3632 


■617 2108 


19 


3899| 7565 
7755 •5941501 


6126 


18 


■618 0145 


17 


•5711612 5437 


4166 


16 


54711 9375 


8188 


15 


93311 •595 3314 


•619 ■2-21 1 


14 


•572 3192 


7255 


6^236 


13 


7054 


•5961196 


•620 0263 


12 


•573 0918 


6140 


4291 


11 


4783 


9084 


8320 


10 


8649 


•597 3030 


•G21 2351 


^ 


•574 2516 


6978 


6.'383 


3 


6385 


•598 0926 


•622 0417 


7 


•575 0-255 


4877 


4452 


6 


41261 882S 


8488 


5 


7999j •599 2781 


•623 25-27 


4 


•576 1873 6735 


6566 


3 


57 48, •000 0691 


•624 0607 


2 


9625 


4648 


4650 


1 


■577 3503 


8606 


8694 





60° 


^d^ 


08« 


/ 



VAT' OOTAW, 



260 



NATURAL TANGENTS. 



32° 

624 8694 

625 2739 
6786 

626 0834 
4884 
8935 

627 2 
7042 

628 1098 
5155 
9214 

629 3274 
7336 

•630 1399 
5464 
9530 

■631 3598 
7667 

■6321 
5810 
9883 

•633 3959 
8035 

•634 2113 
619 

.635 0274 
435 
8441 

•636 2527 
6614 

•637 0703 
4793 



7073 
•639 1169 

5267 

9366 
•640 3467 

7569 
•641 1673 

5779 

9886 
•642 3994 

SI 
•643 -2216 

6329 
.6440444 

4560 

8678 
•645 2797 

6918 
•646 1041 

5165 

9290 
•647 3417 

7546 
•648 1676 

5808 

9941 
•649 4076 

57° 



33° 

649 4076 
8212 

650 2350 
6490 

651 0631 
4774 
8918 

652 3064 
7211 

■653 13601 

5511 

■ 9663 

■654 3817 

7972 
•6G5.2129 

6287 
•656 0447 

4609 

8772 
•657 293' 

7103 
•6581271 

5441 

9612 
•C59 3785 

7960 
•GGO 2136 

6313 
•6C1 0492 

4673 

8856 
•662 3040 

7225 
•663 1413 

5601 

9792 
•664 3984 

8178 
•665 2373 

6570 
•666 0769 

4969 

9171 
•667 3374 

7580 
•6681786 

5995 
•669 0205 

4417 

8630 
•670 2845 

7061 
•671 1280 

5500 

9721 
•672 3944 

8169 
•673 2396 

6624 
•674 0854 

5085 

56° 



34° 

674 5085 
9318 

675 3553 
7790 

676 2028 
6268 

677 0509 
4752 
8997 

678 3243 
7492 

679 1741 
5993 

680 0246 
4501 
8758 

■681 3016 
7276 

■682 153' 
5801 

■683 0066 
4333 
8601 

■684 2871 
7143 

•6851416 



•686 4247 

8528 
•687 2810 

7093 
■6881379 

5666 

9955 
•689 424 

8538 
•690 2S32 

7128 
•691 1425 

5725 
692 0026 

4328 

8633 
.693 2939 

7247 
•6941557 

586S 
■695 0181 

449( 

881f 
.696 3131 

7451 
•697 1773 

6097 
•698 0422 

4749 

9078 
•699 3409 

7741 
•700 2075 

55° 



35° 


36° 


' 37° ! 


700 2075 


726 5425 


•753 5541 • 


6411 


1.871 


•754 0102 


701 0749 


727 4318 


4666 • 


5089 


8767 


9232 


9430 


•728 3218 


•755 3799 


702 3773 


7671 


8369 


8118 


•729 2125 


•756 2941 


703 2464 


6582 


7514 


6813 


•730 1041 


•757 2090 


704 1163 


5501 


6668 


5515 


9963 


•758 1248 


9869 


•731 4428 


5829 


705 4224 


S8'D4 


•759 0413 


8581 


■732 33G2 


4999 


7062940 


7832 


9587 


7301 


•733 2305 


•7C0 4177 


707 1664 


6777 


8769 


6028 


•7341253 


•TGI 3363 


708 0396 


5730 


7959 


4763 


•735 0210 


•762 2557 


9133 


4691 


7157 


709 3504 


9174 


•7631759 


787? 


•736 3G60 


6363 


710 2253 


8147 


•764 0969 


6630 


•737 2636 


5577 


711 1009 


7127 


•7C5 01S8 


5393 


■738 1G20 


4800 


9772 


6115 


9414 


712 4157 


739 OCll 


7GG4a31 


8543 


5110 


SG49 


713 2931 


9G11 


•7G7 3270 


7320 


■740 4113 


7893 


714171? 


8618 


•768 2-517 


610C 


■741 3124 


7144 


715 0501 


7633 


•7691773 


4S9S 


■742 2143 


6404 


9297 


6655 


■770 1037 


716 369? 


•7431170 


5672 


8100 


5686 


•771 0309 


•717 2505 


•7410204 


4948 


6911 


4724 


9589 


•7181319 


9246 


•772 4233 


5729 


•745 3770 


8878 


•719 0141 


8296 


■773 3526 


4554 


•746 2824 


8176 


8970 


7354 


■774 2827 


•720 3387 


•747 1886 


7481 


78or 


6420 


•775 2137 


•721 2227 


■748 0956 


6795 


665C 


5494 


•776 1455 


7221075 


•749 0033 


6118 


550-.' 


4575 


•777 0782 


9930 


9119 


5448 


•723 4361 


750 3665 


•778 0117 


8793 


8-212 


4788 


•724 3227 


■751 2762 


9460 


7663 


7314 


•779 4135 


•725 2101 


■7521867 


8812 


6540 


6423 


■780 3492 


•726 0982 


•753 0981 


8173 


5425 


■ 5541 


•781 2856 


54° 


53° 


52° 



38° 


39° 


r 


781 2856 


809 7840 


60 


7542 


810 2658 


59 


782 2229 


7478 


58 


6919 


811 2300 


57 


•7831611 


7124 


56 


6305 


■812 1951 


55 


■784 1002 


6780 


54 


5700 


■813 1611 


53 


■785 0400 


6444 


52 


5103 


•814 1280 


51 


9808 


6118 


50 


■786 4515 


•815 0958 


49 


9224 


5801 


48 


■787 3935 


816 0646 


47 


8049 


5493 


46 


■788 3364 


•817 0343 


45 


8082 


5195 


44 


•789 2802 


•818 0049 


43 


7524 


4905 


42 


•790 2248 


9764 


41 


6975 


•819 4625 


40 


■791 1703 


9488 


39 


6434 


■820 4354 


38 


■7921167 


9222 


37 


5902 


•821 4093 


36 


•793 0640 


8965 


35 


5379 


•8-22 3840 


34 


•794 0121 


8718 


33 


4865 


.823 3597 


32 


9611 


8479 


31 


■795 4359 


■824 3364 


30 


9110 


8251 


29 


■796 3862 


■825 3140 


.28 


8617 


8031 


27 


■797 3374 


■826 2925 


26 


8134 


7821 


2.^ 


■798 2895 


•827 2719 


•24 


7659 


7620 


23 


•799 2425 


■828 2523 


22 


7193 


7429 


21 


■8001963 


■829 2337 


20 


6736 


7247 


19 


•8011511 


■830 2160 


18 


6288 


7075 


17 


•802 1067 


•831 1992 


16 


5849 


6912 


15 


■803 0632 


•832 18«4 


14 


5418 


6759 


13 


■804 0-206 


•833 1686 


12 


4997 


6615 


11 


9790 


•8341547 


10 


•805 4584 


6481 


9 


9382 


■835 1418 


8 


■806 4181 


6357 


7 


8983 


■836 1-298 


6 


•807 3787 


6-242 


5 


8593 


■837 1188 


4 


•808 3401 


6136 


3 


8-212 


■838 1087 


2 


•809 3025 


6041 


1 


7840 


■839 0996 





51° 


50° 


/ 



NAT. COTAN. 



NATURAL TANGENTS. 



261 



,' 


40° 


41° 


42° 


43° 


44° 


45° 


4G° 


47° 


/ 


3 


839 0990 


869 2867 


•900 4040 


932 5151 


•965 6888 


1^00 00000 


1^03 55303 


V-Q- 23687 60 


1 


595o 


7970 


9309 


933 C591 


•96G2511 


05819 


61333 


29943 59 




840 0915 


870 3087 


•9014580 


6034 


8137 


11642 


67367 


36203168 


3 


5878 


8200 


9854 


934 1479 


•967 3767 


17469 


73404 


42467157 


4 


841 0844 


871 3316 


•902 5131 


6928 


9399 


23298 


79445 


48734 56 


5 


5812 


8435 


•903 0411 


935 2380 


•968 5035 


29131 


85489 


55006 65 


6 


842 0782 


872 355G 


5! 95 


7834 


•969 0074 


34968 


91538 


61282164 


7 


5755 


8r8C 


•904 007'.: 


936 3292 


6316 


40807 


97589 


67561153 


8 


•8430730 


873 3S0G 


6207 


8753 


•9701962 


46051 


1^04 03645 


73845162 


9 


6708 


8935 


•905 1557 


937 4216 


7610 


5-2497 


09704 


80132 51 


10 


•844068S 


8744067 


6851 


9383 


•971 3262 


58348 


15767 


86423 60 


11 


5C70 


9201 


•90G2147 


938 5153 


8917 


64201 


21833 


92718 49 


12 


•8450C55 


875 433S 


7440 


939 0625 


•972 4575 


70058 


27904 


99018 48 


13 


5643 


947? 


•907 2748 


6101 


•973 0-230 


75918 


33977 


V08 05321 47 


14 


•846 0C33 


8764020 


8053 


940 1579 


5901 


81782 


40055 


11628,46 


15 


5C25 


9765 


•903 3300 


7061 


•9741569 


87649 


46136 


1793945 


16 


•847 0C20 


877 4012 


8G71 


941 2545 


7240 


93520 


52221 


24'254!44 


17 


5G17 


878 0062 


•909 3984 


8033 


•975 2914 


99394 


58310 


30573143 


18 


•848 0C17 


5215 


9300 


942 3523 


8591 


1-01 05272 


64402 


36896142 


19 


5G19 


879 0370 


•910 4619 


9017 


•976 4-272 


11153 


70498 


43223 41 


20 


•849 0C24 


5528 


9940 


•943 4513 


9956 


17038 


76598 


49554 40 


21 


5031 


880 0:8? 


•911 5265 


•944 C013 


•977 5643 


22925 


82702 


55889 J39 


22 


•850 0640 


5C52 


•912 0592 


5516 


•978 1333 


28817 


88809 


62228 [38 


23 


5C53 


881 1017 


5922 


■945 1021 


7027 


34712 


94920 


6857137 


24 


•851 0CC7 


61 86 


•9131255 


6530 


•979 2724 


40610 


r05 01034 


74918136 


25 


5C84 


•882 1357 


6591 


•946 2042 


8424 


46512 


07153 


812'J9i36 


26 


•852 0704 


6531 


•914 1929 


7556 


•980 4127 


5-2418 


13275 


87624 34 


27 


572L 


•8831707 


7270 


•947 3074 


9833 


58326 


19401 


9.-.984I33 


28 


•853 075: 


6886 


•915 2015 


8595 


•981 5543 


64239 


25531 


1-09 00347132 


23 


5777 


•884 2068 


7962 


•948 4119 


•9821256 


70155 


31664 


06714 31 


30 


•8540807 


7253 


•916 3312 


9646 


6973 


76074 


37801 


13085 


30 


31 


5S39 


•885 2440 


8665 


•949 5176 


•983 2692 


81997 


43942 


19460 


29 


32 


•855 0873 


7030 


•917 4020 


•950 0709 


8415 


87923 


50087 


26840 


28 


33 


5910 


•8862822 


9379 


6245 


•9844141 


93853 


56235 


32223 


27 


34 


•856 0950 


8017 


•918 4740 


•951 1784 


9871 


99786 


623F-8 


38610 


26 


35 


5992 


•887 3215 


•919 0104 


7326 


•985 5603 


1-02 05723 


68.544 


45002 


25 


36 


•857 1037 


8415 


5471 


•952 2871 


•986 1339 


11664 


74704 


51397 


24 


37 


6084 


•888 3619 


•920 0841 


8420 


7079 


17C08 


80867 


57797 


23 


3S 


•858 1133 


8825 


6214 


•953 3971 


•987 2821 


23555 


87035 


64201 


22 


39 


6185 


•889 4033 


•921 1590 


9526 


8567 


29506 


93206 


70609 


21 


40 


•859 1240 


9244 


6969 


•954 5083 


•988 4316 


35461 


99381 


77020 


20 


41 


6297 


•890 415? 


•922 2350 


•955 0644 


•989 0069 


41419 


1^06 05560 


83436 


19 


42 


•860 1357 


9675 


7734 


6208 


5825 


■ 47381 


11742 


89857 


18 


43 


6419 


•891 4894 


•923 3122 


•9561774 


•990 1584 


53346 


17929 


96281 


17 


44 


•861 1484 


•892 0116 


8512 


7344 


7346 


59315 


24119 


MO 02709 


16 


45 


6551 


5341 


•924 3905 


•957 2917 


•991 3112 


65287 


30313 


09141 


15 


46 


•8621621 


•893 0569 


9301 


8494 


8881 


71263 


36511 


15578 


14 


47 


6694 


5799 


•925 4700 


•958 4073 


•992 4054 


77243 


42713 


22019 


13 


48 


•8631768 


•8941032 


•926 0102 


9655 


•993 0429 


83226 


48918 


28463 12 


49 


6846 


6268 


5506 


•959 5241 


6-208 


89212 


55128 


3491211 


50 


•864 1926 


•895 1506 


•927 0914 


•960 0829 


•9941991 


95203 


61341 


41365 |l0 


51 


7009 


6747 


6324 


6421 


7777 


1-03 01196 


67558 


47823 


9 


52 


•865 2094 


•8961991 


•9281738 


•961 2016 


•995 3566 


07194 


73779 


54284 


8 


53 


7181 


7238 


7154 


7614 


9358 


13195 


80004 


60750 


7 


54 


•866 2277 


•897 2487 


•929 2573 


•962 3215 


•996 5154 


19199 


86233 


67219 


6 


55 


736b 


7739 


7996 


8819 


•997 0953 


25208 


9246f^ 


73693 


5 


56 


•867 246G 


•898 2994 


•930 3421 


•963 4427 


6756 


31220 


98702 


801711 4 


57 


7558 


8251 


8849 


•964 0037 


•998 2562 


37235 


1^07 04943 


86663 3 


58 


•868 2659 


•899 3512 


•931 4280 


5651 


8371 


43254 


11187 


93140 2 


59 


7762 


8775 


9714 


•965 1268 


.999 4184 


49277 


17435 


996301 1 


60 


•869 2867 


•900 4040 


•932 5151 


6888 


l^OOOOOOO 


55303 


23687 


Ml 061251 


/ 


49° 


48° 


47° 


46° 


45° 


44° 


43° 


42° 


' 



SAT, COZUb 



202 



NATURAL TANGENTS. 



f 


4»-.' 


49° 


50° 


51° 


52° 


53° 


54° 


/ 





Ill 06125 


115 03684 


119 17536 


1-23 48972 


1-27 99416 1-32 7044<, 


1-37 63819 


60 


1 


12,624 


10445 


24579 


56319 


1-28 07094 


78483 


72242 


59 




19127 


17210 


31626 


63672 


14776 


86524 


80672 


58 


3 


25635 


23979 


38679 


71030 


2-2465 


94571 


89108 


57 


4 


32 U6 


30754 


45736 


78393 


30160 1-33 026:^4 


97551 


56 


5 


386(32 


37532 


52799 


85762 


37860 


10684 


1-38 06001 


55 


f 


■ 45182 


44316 


59866 


93136 


45566 


18750 


14458 54 


7 


5i70() 


51104 


66938 


1-24 00515 


53-277 


26822 


22922 53 


8 


582:J5 


57896 


74015 


07900 


60995 


34900 


31392 52 


9 


W7G8 


64693 


81097 


15-290 


08718 


42984 


39869 51 


10 


71305 


71495 


88184 


22685 


76447 


51075 


48353 


50 


11 


77840 


78301 


95276 


30080 


84182 


59172 


56844 


49 


12 


84391 


85112 


1-20 0-2373 


37492 


91922 


67276 


65342 


48 


13 


90941 


91927 


09475 


44903 


99669 


75386 


73847 


47 


14 


97495 


98747 


16581 


52320 


129 07421 


83502 


82358 


46 


15 


1-12 04053 


1-16 05571 


23693 


59742 


15179 


91624 


90876 


4£ 


16 


10616 


1-2400 


30810 


67169 


22943 


99753 


99401 


44 


17 


17183 


19384 


37932 


74602 


30713 1-34 07888 


1-3907934 


43 


18 


23754 


26073 


45058 


82040 


38488 


16029 


16473 


42 


19 


30329 


32916 


52190 


89484 


46270 


24177 


25019 


41 


20 


36909 


39763 


59327 


96933 


54057 


3-2331 


33571 


40 


21 


43493 


46615 


6f>468 


1-25 04388 


61850 


40492 


4-2131 


39 


22 


50081 


53472 


73615 


11848 


69649 


48658 


50698 


38 


23 


5G674 


60334 


80767 


19313 


77454 


56832 


59272 


37 


24 


63271 


67200 


87924 


26784 


85265 


65011 


67852 


36 


25 


69872 


74071 


95085 


34260 


93081 


73198 


76440 


35 


26 


7(478 


80347 


1-21 02252 


41742 


1-30 00904 


81330 


85034 


31 


27 


83088 


87827 


094-24 


49229 


08733 


895S9 


93636 


33 


28 


£9702 


94712 


16601 


56721 


16567 


97794 


1-40 02245 


32 


29 


P0321 


1-17 01601 


23783 


64219 


24407 


1-35 06006 


10860 


31 


30 


1-1302944 


08496 


30970 


71723 


32254 


14224 


19483 


30 


31 


09571 


15395 


38162 


79-232 


40106 


22449 


•28113 


29 


32 


16203 


2 2238 


45359 


86747 


47964 


30680 


36749 


28 


33 


22839 


23207 


52562 


94267 


55828 


38918 


45393 


27 


34 


29479 


36120 


59769 


1-26 01792 


63699 


47162 


54044 


26 


35 


3612 i 


43038 


66982 


09323 


71575 


55413 


6'2702 


,25 


36 


42773 


49960 


74199 


1CS60 


79457 


63670 


71367 


24 


37 


49427 


56.SS8 


81422 


24402 


87345 


71934 


80039 


23 


38 


56085 


6382) 


88650 


31950 


95-239 


80204 


88718 


22 


39 


62747 


70756 


95883 


39503 


1-3103140 


88481 


97406 


21 


40 


69414 


7769S 


1-2203121 


47062 


11046 


96764 


1-41 06098 


20 


41 


76086 


84644 


10364 


54626 


18958 


1-36 05054 


14799 


19 


42 


82761 


91595 


17613 


62196 


26876 


13350 


23506 


IS 


43 


89441 


98551 


24866 


69772 


34801 


21653 


32221 


17 


44 


96126 


■1-18 05512 


32125 


77353 


42731 


29963 


40943 


16 


45 


1-U02815 


1-2477 


39389 


84940 


50668 


38-279 


49673 


15 


46 


09508 


19447 


46658 


92532 


58610 


46602 


5840C. 


14 


47 


16206 


26422 


53932 


1-27 00130 


66559 


54931 


67153 


13 


48 


22908 


33402 


61211 


07733 


74513 


63267 


75904 


12 


49 


29615 


40387 


68496 


15342 


82474 


71610 


84662 


11 


60 


363^1 


47376 


75786 


22957 


90441 


79959 


03427 


10 


61 


430 U 


54370 


83081 


30578 


98414 


883i5 


r42 02200 


9 


52 


49762 


C1369 


90381 


38204 


1-32 06393 


96678 


10979 


8 


53 


56486 


68373 


97687 


45835 


14379 


1-37 05047 


19760 


7 


54 


63215 


75382 


1-2304997 


53473 


22370 


13423 


28561 


6 


55 


69949 


82395 


12313 


61116 


30368 


•21800 


37362 


5 


5o 


76687 


89414 


19634 


68765 


38371 


30195 


46171 


4 


87 


83429 


96437 


26961 


76419 


40381 


38591 


54988 


3 


58 


90176 


1-19 03465 


34292 


84079 


54397 


4G994 


63811 


2 


59 


96928 


10498 


41629 


91745 


62420 


55403 


7'2642 


1 


60 


1 -la 03684 


17536 


48972 


99416 


70448 


63819 


81480 





/ 


410 


40° 


39° 


38° 
AT. COTi 


37° 


36° 


36° 


/ 



NATUBAL TANGENTS. 



26:3 



/ 


55° 


56° 


57° 


58° 


59° 


60° 


61° 


/ 





1-42 81480 


L-4S 25610 


l-:3 9S650 


1-60 03345 


1-66 4-2795 


1-73 20508 


1-80 40478 


CO 


1 


90320 


34916 


1-54 08-160 


13709 


537 06 


32149 


52860 


59 


2 


99178 


44231 


18280 


24082 


C4748 


43803 


C:)25G 


58 


3 


1-43 08039 


53554 


2S10S 


34465 


75741 


55468 


77664 


57 


4 


1G90C 


62SS4 


37940 


44858 


S6744 


07144 


00080 


56 


5 


257 SI 


72223 


47792 


55260 


97758 


78833 ' 


1-61 U2521 


55 


6 


34604 


81570 


57647 


65672 


1-67 08782 


90533 


14969 


54 


7 


43554 


90925 


07510 


76094 


19818 


1-7402245 


27430 


53 


8 


52451 


L-49 00288 


77383 


80525 


30804 


13969 


39904 


52 


y 


61356 


09659 


87264 


96966 


41921 


25705 


52391 


51 


10 


70268 


19039 


97155 


1-61 07417 


52988 


37453 


64892 


50 


11 


79187 


28426 


1-55 07054 


17878 


64067 


49213 


77405 


49 


12 


88114 


37822 


16963 


28349 


•^5150 


60984 


89932 


48 


13 


97049 


47225 


26880 


38829 


86250 


72768 


1-82 02473 


47 


14 


L-44 05991 


56637 


36806 


49320 


973C7 


84564 


15026 


40 


15 


14940 


6605S 


46741 


59820 


1-68 08489 


96371 


27593 


45 


16 


23897 


75486 


56685 


70330 


19621 


1-75 08191 


40173 


44 


17 


3286:: 


84923 


06639 


80850 


307C5 


20023 


52767 


43 


18 


4183i 


943G7 


76601 


91380 


41919 


31860 


65374 


42 


19 


50814 


1-50 03821 


86572 


1-62 01920 


53085 


43722 


77994 


41 


20 


59801 


13282 


96552 


12469 


64261 


55590 


90628 


40 


21 


68796 


22751 


1-56 06542 


23029 


75449 


67470 


1-83 03275 


39 


22 


77798 


3-2229 


16540 


33599 


86647 


79362 


15936 


38 


23 


86808 


417ie 


26548 


44178 


97856 


91267 


28610 


37 


U 


95825 


51210 


36564 


54768 


1-69 09077 


1-7603183 


41297 


36 


lb 


1-45 04850 


60713 


46590 


65368 


20308 


15112 


53999 


35 


26 


13883 


702-24 


56625 


75977 


31550 


27053 


66713 


34 


27 


22923 


79743 


66669 


86597 


42804 


39007 


79442 


3D 


28 


31971 


89271 


76722 


97227 


54069 


50972 


92184 


32 


29 


41027 


98807 


86784 


1-63 07867 


65344 


62950 


1-8404940 


31 


30 


50090 


1-51 08352 


96856 


18517 


76631 


74940 


17709 


30 


31 


59161 


17905 


1-57 06936 


29177 


87929 


86943 


30492 


29 


32 


68240 


27466 


17026 


39847 


99238 


98958 


43289 


28 


33 


77326 


37036 


27128 


50528 


1-70 10559 


1-77 10985 


56099 


27 


34 


86420 


46614 


37234 


61218 


21890 


23024 


68923 


26 


35 


95522 


56201 


47352 


71919 


33233 


35076 


81761 


25 


36 


1-46 04632 


65796 


57479 


82630 


44587 


47141 


94613 


24 


37 


13749 


75400 


67615 


93351 


55953 


59218 


1-85 07479 


23 


38 


22874 


85012 


77760 


1-64 04082 


67329 


71307 


20358 


23 


39 


32007 


94632 


87915 


14824 


78717 


83409 


33252 


21 


40 


41147 


1-52 04261 


98079 


25576 


90116 


95524 


46159 


20 


41 


50296 


13899 


1-58 08253 


36338 


1-71 01527 


1-78 07651 


59080 


19 


42 


59452 


23545 


18436 


47111 


12949 


19790 


72015 


18 


43 


68616 


33200 


28628 


57893 


24382 


31943 


84965 


17 


44 


77788 


42863 


38830 


68687 


35827 


44107 


97928 


16 


45 


86967 


52535 


49041 


79490 


47283 


56285 


1-8610905 


1& 


46 


96155 


62215 


59261 


90304 


58751 


68475 


1 23896 


14 


47 


1-47 05350 


71904 


69491 


1-65 01128 


70230 


80678 


36902 


13 


48 


14553 


81602 


79731 


11963 


81720 


92893 


49921 


12 


49 


23764 


91308 


89979 


22808 


93222 


1-79 05121 


62955 


11 


50 


32983 


1-53 01023 


1-59 00238 


33663 


1-72 04736 


17362 


76003 


10 


51 


42210 


10746 


10505 


44529 


16261 


29616 


89065 


9 


52 


51445 


20479 


20783 


55405 


27797 


41883 


1-87 02141 


8 


53 


606S8 


30219 


31070 


66292 


39346 


54162 


15231 


T 


54 


69938 


39969 


41366 


77189 


50905 


66454 


28336 


6 


55 


79197 


49727 


51672 


88097 


62477 


78759 


41455 


5 


56 


88463 


59494 


61987 


99016 


74060 


91077 


54588 


4 


57 


97738 


6927C 


72312 


1-66 09945 


85654 


1-80 03408 


67736 


3 


58 1-48 07021 


79054 


82647 


20884 


97260 


15751 


80898 


2 


59 16311 


8884S 


9-2991 


31834 


1-73 08878 


28108 


94074 


1 


60 25610 


9865C 


1-60 0334£ 


\ 42795 


20508 


40478 


1-88 07266 





/ 340 


33° 


32° 


31° 


f 30° 


29° 


28° 


' 








J 


iAX. COT 


iN 









'^64 



NATURAL TANGENTS. 



' 1 


62° 


63° 


64° 


65° 


66° 


67° 


68° 


/ 





1-88 07265 


1-96 -26105 


^•05 03038 


2-1445069 


2-24 60368 


2-35 58524 2-47 50869 


60 


1 


20470 


41)227 


18185 


61366 


77962 


77590 


71612 


59 


2 


33C0G 


54314 


3334U 


77b83 


955S0 


96683 92386 


58 


3 


46924 


68518 


48531 


94021 


2-25 13221 


2-3615801 


2-48 13190 


57 


4 


00172 


82688 


63732 


2-15 1037S 


30885 


34946 


^4023 


56 


5 


73436 


96874 


78950 


•26757 


48572 


54118 


54887 


55 


6 


86713 


1-97 11077 


94187 


43156 


60283 


73310 


75781 


64 


7 


1-89 00006 


25296 


2-06 09442 


59575 


84016 


92540 


90706 


53 


8 


13313 


39531 


24716 


76015 


2-20 01773 


2-37 11791 


2-49 17660 


52 


9 


26635 


53782 


40008 


92476 


19554 


31068 


38645 


51 


10 


39971 


68050 


55318 


2-16 08958 


37357 


50372 


59661 


60 


11 


53322 


82334 


70646 


25460 


55184 


69703 


80707 


49 


12 


66688 


96635 


85994 


41983 


73035 


89060 


2-50 01784 


48 


13 


80068 


1-98 10952 


2-07 01359 


58527 


90909 


2 38 08444 


22891 


47 


14 


934C4 


25286 


16743 


75091 


2-27 0S807 


27855 


44029 


46 


15 


1-90 06874 


39636 


32146 


91677 


26729 


47293 


65198 


45 


16 


20299 


54003 


47567 


2-17 08283 


44674 


66758 


86398 


44 


17 


33738 


68387 


63007 


24911 


62643 


86250 


2-5107629 


43 


18 


47193 


82787 


78465 


41559 


80636 


2-39 05769 


28890 


42 


19 


60G63 


97204 


93942 


58229 


9S653 


25316 


50183 


41 


20 


74147 


1-99 11637 


2-0809438 


74920 


2-28 16693 


44889 


71507 


40 


21 


87647 


26087 


24953 


91631 


34758 


64490 


92863 


39 


22 


1-91 01162 


40554 


40487 


2-18 08364 


52846 


84118 


2-52 14249 


38 


23 


14691 


55038 


56039 


25119 


70959 


2-40 03774 


35667 i 37 


24 


28236 


69539 


71610 


41894 


89096 


23457 


57117 1 36 


25 


41795 


84056 


87200 


, 58691 


-2-29 07257 


43168 


78598 ! 35 


26 


55370 


98590 


2-09 02809 


75510 


25442 


62906 


2-53 00111 ' 34 


27 


68960 


2-00 13142 


18437 


92349 


43651 


82672 


21655 ' 33 


28 


82565 


27710 


34085 


2-19 09210 


61885 


2-41 02465 


43231 i 32 


29 


96186 


42-295 


49751 


26093 


80143 


22286 


64839 31 


30 


1-92 09821 


56897 


65436 


42997 


98425 


42136 


86479 i 30 


31 


23472 


71516 


81140 


59923 


2-30 16732 


62013 


2-54 08151 29 


32 


37138 


86153 


96864 


76871 


35064 


81918 


29855 ' 28 


33 


50819 


2-01 00806 


210 12607 


93840 


53420 


2-4201851 


51591 i 27 


34 


64516 


15477 


441 5C 


2-20 10831 


71801 


21812 


73359' 26 


35 


78228 


30164 


27843 


90206 


41801 


95160; 25 


36 


91956 


44869 


59951 


44878 


2-31 08637 


61819 


2-55 16992 I ^ 


37 


1-93 05699 


59592 


75771 


61934 


27092 


81864 


38858' 23 


38 


19457 


74331 


91611 


79012 


45571 


2-43 01938 


607561 22 


39 


33231 


89088 


2-11 07470 


96112 


64076 


22041 


82686 21 


40 


47020 


2-02 03862 


23848'2-21 13234 


82606 


42172 


2-56 04649! 20 


41 


60825 


18654 


3924e 


30379 


2-32 01160 


62331 


26645! 19 


42 


74545 


33462 


55164 


47545 


19740 


82519 


486741 J8 


43 


88481 


48289 


71101 


64733 


38345 


2-44 02736 


70735 ' 17 


44 


1-9402333 


63133 


87057 


81944 


56975 


22982 


92830' 16 


45 


16200 


77994 


2-12 03034' 99177 


75630 


43256 


2-57 14957 i 15 


46 


30083 


92873 


19030 2-22 16432 


9431 1 


63559 


37118 • 14 


47 


43981 


2-03 07769 


35046 33709 


2-3313017 


83891 


59312 1 J3 


48 


57896 


22683 


51082 51009 


31748 


2 45 04252 


81539 12 


49 


7182e 


37615 


671371 68331 


50505 


24642 


2-58 03S00 11 


50 


85772 


52565 


83-213 85676 


69287 


45061 


26094 10 


51 


9973C 


67532 


993082-23 03043 


88095 


65510 


48421 9 


52 


1-9513711 


82517 


2-1315423 20433 


2-34 06928 


85987 


70782 8 


53 


27704 


97519 


31559 37845 


25787 


2.46 06494 


9?177 , 7 


54 


4171^ 


2-04 12540 


47714 55280 


44672 


27030 


2-59 15606 6 


55 


5573( 


27578 


63890 72738 


63582 


47596 


38068 5 


56 


6978( 


) 42534 


80085 90218 


8-2519 


68191 


60564 4 


57 


8883' 


57708 


963012-24 07721 


2-35 01481 


88816 


83095 3 


58 


9791( 


) 7280C 


2-1412537 25247 


20469 


-:-47 09470 


2-60 05659 2 


59 


1-961200 


) 8791( 


28793 42796 


■ 39483 


30155 


28258 1 


60 


2610512-05 0303J 


45069, 60368 


58524 


50869 


508911 


• 


27° 


1 26° 


25° 


i 24° 


23° 


22° 


21° 


1 / 



VAX. «OTAJS. 



NATURAL TANGENTS. 



f 


69° 


70° 


71° 







2-60 50891 


2-74 74774 


■2-90 42109 


3- 


1 


73558 


99661 


69576 


3- 


2 


96259 


2-75 24588 


97089 




3 


2-61 18995 


49554 


2-91 24649 




4 


41766 


74561 


5225C 




5 


64671 


99608 


79909 


3- 


6 


87411 


2-76 24685 


2-9207010 




7 


2-62 10286 


49822 


35358 




8 


33196 


74990 


63152 


3- 


9 


56141 


2-77 00199 


90995 




10 


79121 


25-US 


■2-9318885 




11 


2-6302136 


50738 


46822 


3- 


12 


25186 


76069 


74807 




13 


48271 


2-78 01440 


2-94 02840 




14 


71392 


26853 


30921 


3- 


15 


94549 


52307 






16 


2-64 17741 


77802 


87227 




17 


40969 


2-79 03339 


2-95 15463 


3- 


18 


64232 


28917 


43727 




19 


87531 


54537 


72050 




20 


2-65 10867 


80198 


•2-96 00422 




21 


34238 


-2-80 05901 


28842 


3- 


22 


57&15 


31646 


57312 




23 


81089 


57433 


85831 




24 


2-66 04569 


83-263 


2-97 14399 


:> 


25 


28085 


2-81 09134 


43016 




26 


51638 


35048 


71G83 




27 


75227 


61004 


2-98 00400 


3- 


2,-> 


98853 


S7003 


-291 G7 




2.1 


2-67 22516 


2-8213045 


57983 




30 


46215 


39129 


86850 


■->- 


:51 


69951 


65256 


2-99 15766 




?>l 


93725 


91426 


44734 




'■'>■) 


2-68 17535 


2-83 17639 


73751 


3- 


\n 


413S3 


43S96 


3-00 02820 




3'. 


652G7 


70196 


31939 




36 


89190 


96539 


61109 


3- 


37 


2-69 13149 


2-84 229-26 


90330 




38 


37147 


49356 


3-01 10603 




33 


61181 


75831 


48923 


3- 


40 


85254 


2-85 02349 


78301 




41 


2-70 093&4 


28911 


3-02 07728 




42 


33513 


55517 


37207 


3- 


43 


57699 


8216S 


06737 




44 


81923 


2-86 08863 


96320 




45 


2-71 06186 


35602 


3-03 25954 


3- 


46 


30487 


62386 


55641 




47 


54826 


89215 


85381 




48 


79204 


■2-8716088 


3-04 15173 


3- 


49 


2-72 036-20 


43007 


45018 




50 


28076 


69970 


74915 




51 


52569 


96979 


3-05 04866 


3- 


52 


77102 


2-88 24033 


34870 




53 


2-73 01674 


51132 


&49-28 




oi 


26284 


78277 


95038 


3- 


55 


50934 


2-89 05467 


3-06 25-203 




56 


756-23 


32704 


55421 




57 


2-74 00352 


5;)9St^ 


85694 


3- 


53 


25120 


87314 


3-07 16020 




59 


499-27 


2-90 146SS 


46400 




6C 


74774 


42109 


76835 


3- 


/ 


2a° 


19° 


18° 





72° 

07 76835 

08 07325 
37869 
68468 
99122 

09 29831 
60596 
91416 

10 22291 
53223 
84210 

11 15254 
46353 
77509 

12 08722 



71317 

13 02701 
34141 
65639 
97194 

14 28807 
60478 
92207 

15 23994 
55S40 
87744 

16 19706 
51728 



73° 

3-27 08526 

42588 
76715 

3-28 10907 
45164 
79487 

3-29 1J^7G 



82851 

3-30 17438 
52091 
86811 

3-31 21598 
56452 
91373 

3-32 26362 
61419 
96543 

3-33 31736 
66997 

3-34 02326 
37724 
73191 

3-35 08728 
44333 
80008 

3-3615753 
51568 
87453 

3-37 23408 



1715948 
48147 I 
804063 

18 12724 
45102 3 
77540 

19 10039 
42598 3' 
75217 

20 07897 



59434 
95531 

38 31699 
67938 

39 04249 
40631 
77085 

40 13612 
50210 
86882 



40638 3-4123626 
73440 60443 
21 0G304! 97333 
39228 3-42 34297 
722151 71334 

22 05233 3-43 08446 
38373] 45631 
7154G, 82891 

23 04780 3-44 20226 
380781 57035 



74° 

3-48 74144 
3-49 12470 

50874 

• 89356 

3-50 27916 

66555 
3-51 05273 

44070 

82946 
3-52 21902 



i-53 00054 
39251 
78528 

1-541788G 
57325 
96840 

;-55 36449 
76133 

;-56 15900 
55749 
95681 

;-57 3569G 
75794 

1-58 15975 
56241 
96590 

;-59 37024 
77543 

.-60 18146 
58835 



714381 

24 04860 3- 
383461 
71895 3- 

25 055081 
391841 
72924 3- 

26 067281 
40596' 
74529 3- 

27 085261 

17° 



95120 I 

45 32679 1 3 
70315 I 

46 08026 3 
45813 
83676 ,3' 

47 21616 
59632 1 
97726 3' 

48 35896 i 
74144 3- 

16° I 



61 40469 
81415 

62 22447 
63566 

6304771 
46064 
87444 

64 28911 
70467 

65 12111 
53844 
95C65 

66 37575 
79575 

67 21665 
63845 

68 06115 
48475 
90927 

69 33469 
76104 

70 18830 
61648 

71 04558 
47561 
90658 

72 33847 
77131 

73 20508 

15° 



76° 

3-73 20508 

63980 
3-7407546 

51207 

94963 
3-75 38815 

82763 
3-76 26807 

70947 
3-77 15185 

59519 
3-78 03951 

48481 

93109 
3-79 37835 

82661 
3-SO 27585 

72609 
3-81 17733 

62957 
3-82 08281 

53707 

99233 
3-83 44861 

90591 
3-84 36424 

82358 
3-85 28396 

745S7 
3-86 20782 

67131 
3-87 13584 

60142 
3-88 06805 

53574 
3-89 00448 

47429 

94516 
3-90 41710 

89011 
3-91 36420 

83937 
3-92 31563 

79297 
-93 27141 

75094 
•94 23157 

71331 
3-95 19615 

68011 
5 16518 

65137 
3-97 13868 

62712 
3-98 11669 

60739 
3-99 09924 

59223 
4-00 08636 

58165 
4-01 07809 

14° 



NAT. COTAIT. 



266 



NATURAL TANGENTS. 



76° 

4-01 07803 

57570 
4-02 074-10 

57440 
4-03 07550 

57779 
4-(>4 081-25 

585J0 
4-05 09174 

59877 
1-06 10700 

61643 
4-07 12707 



4-08 15199 

66627 
4-0918178 

69852 
4-10 21649 

73560 
4-11 2561 i 

777S4 
4-12 30079 

82493 
4-13 3504G 

87719 
4-1440519 

93446 
4-15 46501 

99685 
4-16 52998 
4-17 06440 

60011 
4-18 13713 

67540 
4-19 21510 

75606 
4-20 29835 

84190 
4-21 38690 

Q3318 
4-22 48080 
4-23 02977 

58003 
4-24 13177 

684S2 
4-25 23923 

79501 
4-26 35218 

91072 

27 47066 

4-28 03199 

59472 

4-29 15885 

72440 
4-30 29130 
85974 
4-31 42955 
4-32 00079 
57347 
4-3314759 

13° 



77° 


78° 


79° 


80° 1 


81° I 


82° 1 


/ 


4-33 14759 k 


1-70 46301 


5-1 445540 


5-6 712818 6 31375151 


--1 153697! 


60 


72310 


t-71 13086 


525557 


809440 


2566011 


304190 59 


i-34 30018 


81256 


005813 


906394 


370126 


455308 58 


878t6 ^ 


t-72 49012 


686311 


3-7 003663 


496092 


607056 57 


4-35 45C61 


4-7316354 


7C7051 


101256 


610502 


759437 56 


4-36 04 J03 


.85083 


848035 


199173' 


737359 


91^456 55 


62293 - 


4-74 53401 


9232C4 


2974161 


858G65 


7-206G116 54 


4-37 20731 


4-75 21907 


5-2 010738 


3959881 


980422 


220422 53 


79317 


93603 


092459 


494889 0-4 102G33| 


375378 52 


4-38 38054 - 


4-76 53490 


174428 


594122 


2253011 


5309871 51 


93940 


4-77 28568 


256047 


C93CS8, 


348428 


687255 50 


4-39 55977 


97837 


339110 


793588 


472017 


844184 49 


4-40 15164 


i-78 67300 


421836 


893825! 


596070 


7-3 001780 48 


74504 


4-79 36957 


504809 


994400 


720591 


160047 i 


47 


4-41 33996 


4-80 00808 


588035 


5-8 095315 


845581 


318989: 


46 


93641 


70854 


671517 


190572 


971043 


478610' 


45 


4-42 53439 


4-81 47096 


755255 


298172 6-5 096981 


638916 


44 


4-43 13392 


4-8217536 


839251 


400117 


223396 


799909 


43 


73500 


88174 


923505 


502410 


350293 


961595 


42 


4-44 33702 


4-83 59010 


5-3 008018 


605051 


477672 


7-4123978 


41 


94181 


4-8430045 


092793 


708042 


605538 


287064 


40 


4-45 54V5G 


4-85 01282 


177830 


811386 


733892 


450855 


39 


4-4615489 


72719 


263131 


915084 


862739 


6153571 38 


76379 


4-86 44359 


348696 


5-9 019138 


992080 


780570' 37 


4-47 37428 


4-87 10201 


434527 


123550 6-6121919 


94G514' 36 


9S636 


88248 


520020 


228322 


25-2258 


7-5113178 35 


4-48 60004 


4-88 60499 


600993 


333455 
438952 


383100 


280571 


34 


4-49 21532 


4-89 32956 


693030 


514449 


448699 


33 


83221 


4-90 05620 


780538 


544815 


646307 


617567 


32 


4-50 45072 


78491 


867718 


051045 


778677 


787179 


31 


4-51 07085 


4-91 51570 


955172 


757644 


911562 


957541 


30 


692C1 


4-92 24859 


5-4 042901 


864014 


6-7 044966 


7-6128657 


29 


4-52 31601 


98358 


130906 


971957 


178891 


300533 


28 


94105 


4-93 72068 


219188 


0-0 079676 


313341 


473174 


27 


4-53 56773 


4-94 45990 


307750 


187772 


448318 


646584 


26 


4-54 19608 


4-95 20125 


39G592 


296247 


583826 


820769 


25 


82008 


94474 


485715 


405103 


719867 


995735 


24 


4-55 45776 


4-90 69037 


575121 


514343 


856446 


7-7 171486 


23 


4-50 09111 


4-97 43817 


664812 


623967 


993565 


348028 


22 


72615 


4-98 18813 


7547S8 


733979 


6-8 131227 


525366 


21 


4-57 36287 


94027 


845052 


844381 


269437 


703506 


20 


4-58 00129 


4-99 69459 


935004 


955174 


408190 


882453 


19 


64141 


5-0045111 


5-5 026446 


6-1 066360 


547508 


7-8 062212 


18 


4-59 28325 


5-01 20984 


117579 


177943 


G87378 


' 242790 


17 


92680 


97078 


209005 


289923 


827807 


424191 


16 


4-60 57207 


5-02 73395 


300724 


402303 


968799 


606423 


15 


4-61 21908 


5-03 49935 


392740 


515085 


6-9 110359 


789489 


14 


86783 


5-04 26700 


485052 


628272 


252489 


973396 


13 


4-62 51832 


5-05 03693 


577003 


741S65 


395192 


7-9 158151 


12 


4-63 17050 


80907 


670574 


855867 


538473 


343758 


11 


82457 


5-06 58352 


763786 


970279 


682335 


530224 


10 


4-64 48034 


5-07 3'o025 


857302 


6-2 085106 


826781 


717555 


9 


4-0513788 


5-08 13928 


951121 


200347 


971806 


905756 


8 


79721 


92061 


5-6 045247 


316007 


7-0 117441 


8-0 094835 


7 


4-66 45832 


5-09 70420 


139680 


432086 


263662 


284796 


6 


4-67 121-24 


5-10 490-24 


23-4421 


548588 


410482 


475647 


5 


78595 


5-11 27855 


329474 


665515 


557905 


667394 


4 


4-68 45248 


5-12 00921 


. 4-i4838 


782868 


705934 


860042 


3 


4-09 12083 


862-24 


520516 


900651 


854573 


8-1 053599 


2 


79100 


5-13 65703 


610509 


6-3 018866 


7-1 003826 


248071 


1 


4-70 46301 


5-14 45540 


712818 


137515 


153697 


443464 





Jgo 


W 
Vi 


10° 
VT. OOXAt 


99 
Eft 


go 


r 


/ 



NATURAL TANGENTS. 



f 


83° 


84° ! 


85° 1 


86° 1 


87° 1 


88° 1 


89° 


/ 





3-1 443464 


3-5143645 11.430052 |l4-300e66!l9-081137 |28-636253| 


57-289962 


60 


1 


639786 


410613 


468474 


3606961 


187930 


877089 


58-201174 


59 


2 


837041 


679068 i 


507154 


421230 


295922 


29-122006 


69-265872 


58 


3 


5-2 035239 


949022 


546093 


482273 


405133 


371106 


60-305820 


57 


4 


234384 


9-6 220486 


585294 


543833 


515584 


024499 


61-382905 


56 


6 


434485 


493475 


6'24761 


605916 


027296 


f 82299 


62-499154 55 





c;;5547 


768000 


664495 


6685-29 


740-291 


30-144(19 


63-650741 


54 


■" 


8:>7579 


Q-7 044075 


704500 


731079 


854591 


411680 


64-858008 


53 


8 


S-3 (1405SC 


3-21713 


744779 


795372 


97U219 


6^3307 


00-105473 


62 


9 


244577 


600927 


785333 


859010 


20-087 19 J 


959928 


67-401854 


61 


10 


449558 


381732 


826167 


924417 


205553 


31-241577 


68-750087 


60 


11 


65553C 


9-8 164140 


867282 


989784 


325308 


528392 


70-153346 


49 


12 


862519 


448166 


908682 


15-055723 


446486 


820516 


71-615070 


48 


13 


S-4 070515 


733823 


950370 


122242 


569115 


3-2-11 8099 


73-138991 


47 


14 


279531 


9-9 0-21125 


992349 


189.349 


693220 


421295 


74-729165 


46 


16 


489573 


310088 


12.034622 


2.57052 


818S2S 


730265 


76-390009 


45 


IG 


700051 


600724 


077192 


325358 


945966 


33-045173 


78-126342 


44 


17 


912772 


893050 


120062 


• 394276 


21-074664 


366194 


79-943430 


43 


IS 


8-5 125943 


10018708 


16323G 


463814 


204949 


693509 


81-847041 


42 


19 


340172 


048283 


206716 


533981 


336851 


34-027303 


83-843507 


41 


20 


5554C8 


078031 


250505 


604784 


470401 


367771 


85-939791 


40 


21 


771838 


1079.54 


294609 


07 6233 


605630 


715115 


88-143572 


39 


22 


989290 


138054 


339028 


748337 


742569 


35-069546 


90-463336 


38 


2;-. 


8-6 207833 


108332 


383768 


821105 


881251 


431282 


92-908487 


37 


24 


427475 


198789 


428831 


894545 


22-021710 


800553 


95-489475 


36 


2o 


648223 


229428 


474221 


968667 


163980 


36-177596 


98-217943 


35 


20 


870088 


2r0249 


519942 


16-04.T4S2 


30SC97 


562059 


101-10690 


34 


27 


8-7 093G77 


291255 


.:^65997 


118998 


454096 


950001 


104-17094 


33 


28 


31719: 


.•:22U7 


01-2390 


195225- 


602015 


37-357892 


107-42048 


32 


29 


542461 


353S27 


659125 


272174 


751892 


76r613 


110-89206 


31 


30 


76SS74 


385397 


706205 


349855 


903766 


CS.-1PS459 


114-58865 


30 


31 


990446 


417158 


753034 


428-279 


23-057077 


617738 


118-54018 


29 


32 


8-8 225186 


449112 


801417 


507456 


21306r 


39-056771 


122-77396 


28 


33 


455103 


481261 


849557 


587396 


371777 


505895 


127-32134 


27 


34 


686206 


513607 


898058 


C68112 


532052 


905400 


132-21851 


26 


35 


918505 


646151 


946924 


749614 


694537 


40-435837 


137-50745 


25 


30 


8-9 152009 


578895 


9961 CO 


831915 


859277 


917412 


143-23712 


24 


37 


386726 


611841 


13-045769 


915025 


24-026320 


41-410588 


149-40.-,u-j 


23 


38 


6226C* 


644992 


095757 


998957 


195714 


915790 


156-2.5908 


22 


39 


859843 


678343 


146127 


17-083724 


367509 


42-4S3464 


163-70019 


21 


40 


9-0 098261 


711913 


196883 


169337 


.'^41758 


964077 


171-88540 


20 


41 


337933 


745687 


248031 


255809 


718512 


43-508122 


180-9:^220 


19 


42 


578867 


779073 


299574 


343155 


897826 


44-006113 


190-98419 


18 


43 


821074 


813872 


351518 


431385 


25-079757 


638596 


202-21875 


17 


44 


9-1 064564 


848288 


403867 


520516 


264361 


45-22ri 41 


214-85762 


16 


45 


309348 


882921 


456625 


610559 


451700 


8-2f'3-l 


229-18166 


15 


46 


55543G 


917775 


509799 


701529 


641832 


40-44SS<-2 


246-55198 


14 


47 


802838 


952850 


563391 


793442 


834823 


47-085343 


264-44080 


13 


48 


9-2 051564 


988150 


617409 


886310 


26-0307:^6 


739501 


286-47773 


12 


49 


301627 


11-0-23676 


671856 


980150 


229638 


48-412084 


312-52137 


11 


60 


553035 


059431 


726738 


18-074977 


431600 


49-10?881 


343-77371 


10 


61 


805802 


095416 


782060 


17OS07 


636690 


315726 


381-97099 


9 


62 


9-3 059936 


131635 


837827 


267654 


844984 


.50-548500 


429-71757 


8 


53 


315450 


168089 


894045 


365537 


27-056557 


51-303157 


491-10600 


7 


54 


672355 


•204780 


950719 


464471 


271486 


52-080673 


572-95721 


6 


55 


830663 


^41712 


14-007856 


664473 


489853 


882109 


687-54887 


5 


56 


9-4 090384 


278885 


065469 


665562 


711740 


53-708587 


859-43630 


4 


67 


351531 


316304 


123536 


767754 


937233 


54-561300 


1145-9153 


3 


58 


614116 


353970 


182092 


871068 


28-166422 


55-441517 


1718-87.32 


2 


59 


878149 


391885 


241134 


975523 


399397 


56-350590 


3437-7467 


1 


60 


9.5 143645 


430052 


300666 


19-081137 


636253 


57-289962 


Infinite. 





/ 


6° 


§0 


4° 

9 


3° 
AT. go7^ 


2® 


P 


O*' 


/ 



SLOPES, FOR TOPOGRAPHY. 



TABLE XV 

SLOPES, FOR TOPOGRAPHY. 



1 

1 Degrees. 


Vertical Rise 

in loo 
Horizontal. 


Horizontal 

Distance 

to a Rise of 

10. 


Degrees. 


Vertical Rise 

in TOO 
Horizontal. 


Horizontal 

Distance 

to a Rise of 

10. 


I 


1-75 


572.9 


19 


34.43 


29.0 


2 

1 


3-49 


286.4 


20 


36.40 


27.5 


1 

3 


5-24 


190.8 


21 


38.40 


26.0 


4 


6-99 


143.0 


22 


40.40 


24.7 


5 


8.75 


"4.3 


23 


42.45 


23.5 


6 


10.51 


95.1 


24 


44-52 


22.4 


7 


12.28 


81.4 


25 


46-63 


21.4 


8 


14.05 


71.2 


26 


48.77 


20.5 


9 


15.83 


63.1- 


27 


50.95 


19.6 


lO 


17-63 


56.7 


28 


53.17 


18.8 


II 


19.44 


51.4 


29 


55.43 


18.0 


1 I. 


21.25 


47.0 


30 


57.73 


17.3 


13 


23.09 


43.3 


35 


70.02 


14.. 2 


14 


24-93 


40.1 


40 


83.91 


11.9 


15 


26.79 


37.3 


45 


100.00 


10. 


i6 


28.67 


34.9 


50 


119.17 


8.4 


17 


30.57 


32.7 


55 


142.81 


7.0 


i8 


32.49 


30.7 


60 


173.20 


5.7 





Note.— See page 52, Art. XVHI., for examples in the use of 
Table XV. 



TABLE XVI. 




CHORDS, MIDDLE ORDINATES, EXTERNAL SE- 
CANTS, AND APEX DISTANCES OF A ONE- 
DEGREE CURVE. 

The angles of the table are the 
intersection angles, I, equal to 
the total central angle inchided 
between the tangent points. 

To find the corresponding func- 
tion for any other curve, divide 
the tabular number by the de- 
gree of curvature. 

The unit chord is assumed to 
be one hundred feet long. 

By using radius of 5,730 feet, 
the chord column of the table can be made serviceable for 
plotting. 

To use the table for curves having chords of 20 metres each, 
divide the several tabular functions, that is to say, tbe Chord, 
Ordinate, Ex-sec, and Apex Distance, by 5 times the proposed 
metric degree of curve, for the proper values of said functions 
in metric measure. Thus, for a 2" metric curve, chords 20 
metres each, the divisor would be 2 X 5 = !0; for a 2° 30' 
curve, 12.5, and so on; taking care to reduce minutes to deci- 
mals of a degree before making the multiplication for a divisor 
in each case. 

Also, if the length in metres of any proposed Chord, Ordi- 
nate, Ex-sec or A. D. for a given angle be known, the tabular 
function corresponding to that angle divided by 5 times the 
known chord, ordinate, etc., will ascertain the degree of 
curvature for chords of 20 metres each, using the foregoing 
precaution as to decimals. 

Example 1.-4° curve, intersection angle 48°, 20-metre 
chords, . '. apex dist. = 2551.1 -7-5x4 = 127.55 metres. 

Example 2. — Apex distance 200 metres, angle 58'' 20', .*. de- 
gree of metric curve = 3198 -f- 200 X 5 = 3^^^ degrees = 3° 12', 
chords being 20 metres each. 



270 



FUNCTIONS OF A ONE-DEGREE CURVE. 



2 


o 













n 






n 




S 


S 






























8 


t-. m t^ ro 


t^ 


^S^^ 


8 


1^ 


ro 


8 


t^mo r^roo t->ro 


8 




ro O t^ m 


8 


c 


\o 


ro OvO 














VU 


en Qso m 




HI 










M 








„ 




8 


< 












a^O^O^ O ON 0^ 



u 














































a 




rOOO 






•* 




30 


» 


t^ 


t^oo o 




rovo 


Ov 


N \D 


o 




o 


m 


-lO 


in -^ro 


en 






















inoo o 


m moo 




m^ OO 


M -^ t^ 


N 






1.>1 






ro ro 


(<) 


<1 




■* 






m ir, 




LOVO 


vo 


O vO 


Cv 




t^oooooo 











































O 


O 














r 








» 
o 


CO 


0000 


Tl- 


t-- 


^ 


N 


OO 


oo 


r^ 


t^oo a\ 




cr,\0 


CT. 


N vO 


n 


in M 


\o 


rr, 


•n 


in Tj- ro 


















■*vo 










inoo 




m\c> oo 






N 




(.M 






ro m ro ro (^ 




■* 






in in 




lO^ kO VO VO 


t^ t^ f^ t^OO 00 00 


o 




















o o 














O 


O 


O O 











O 


o 


O 


O 


O 


s 














































Q 


8 




8 




8 








8 




8 








8 




8 




O 


rONO 00 


OS 








o 






rovo 


o 


ro^O 










X 


8 


rovo 


o 


m^o 


n 


rovD 


o 


rO^O 


n 


mvo 


n 


rovD 


s 


r<-)^ 


o 


mvo 


n 


rovO 





mvn OS 








" 








n- 


m 










vo vo 




t^ t^oooooo 


Cy. On ON ON 


:^ 





N TtvO 


00 


o 


M 


^vOCO 


O 




^vOOO 


o 


(M Tt 


vO 00 


N 


T^^O00 


N 


^NDOO 




































T)- 


'I- •<1 










^ 










































' 





O N -^nO 00 O 



-*VO 00 O N -^nO ( 



Tj- -.1- -.^ in in 



Q 

< 


ONo"roONo"roONS' 


roONo"roOND roONO 


roOvo'rnONo'foONO coOno roO 


H m invo CO o w 


m invo 00 w ro invo 00 w rn invo 00 m ro inso 00 




O.OOO 
O.OOO 
O.OOI 
0.002 

0.004 
0.006 
0.009 
0.012 


000000000 
666666666 


6066600 o'odcodd 



t^ 



On D in ON •* ON \ 



N N m 'I- -* inNO t^ I 



00000000000 
0000000006666666666600 



O ■'J-oo 

, ON O H 
M N N 



000000000 



i 










d 


8 
6 


mvo 5 mvo 


m f-, 
rONO 


mNO fONO rOND mso rONO rONO mo O 




a 

CJ 


fONO rONO 


^^^ 


-^0 mNO mNO^ ^NO mvO^ mg mvo 3 


5 





-^NO 00^ 


j^NOOO 


s s ^^^ ^ ^z-^^ % % ?^°^ a ^ ^^^^s 



FUNCTIONS OF A ONE DEGREE CUEVE. 



1 

1 




N T^^oo 


M ^\o 00 N -+VO 00 !N ^^ 00 M -^'O 00 w 'i-vr CO 
MMr,H(NOcin(Nmmmmm-r'*-'j- 1- tc i/^ lo m u, ir,v; 


0- 


t^ Ti- M 00 lA fi 

t-- 1- t^ 'S- 


0\0 m a\vo mo t-^ ^ m o^d mo ^^'^0 t-~M-Moo mw 
t^-TTO i-.TfM t^-^>H r.,mo t^^O t-^O 1^ ^0 t^ ■*• 


w roiovooo O M mu-)vooo o w m iomd oo O " ro in^ oo m ro lo^o oo O 
in vn in u-> in mvo •o^oyD'Ovo t>t^t^t^r>- t^oo 0003000000 osoooooo 


„„„M«MMMM«„N 


CO 


1^8 Jg^HHS^^SSSi^^^cgi H^g^S^HE,'^^! 


M N n 0) (N (N 


(NOiNWf)f)(NNP)NiN(NiNN(NCjmf)mmmmmmm 


2 
Q 


-)-oo rooo moo 
vD in ^^ -too 

M M 


in N n^^ -* m m Pi n n -^vO ooomt-^fivowvomot^'*-" 
mob (Nti^tvJtN i^D i^Pi r^NOT moo ^fOMnO^ n r^mo> 
N (N <r,m-i- -f \n in\o vo t^ r~.oo ooo^o^OOwP^lNmm-^•il- 


H N N N N N 


Pl(N(N(N(N<NNNC4N!NNNNNP)mmmmmmmrOff> 


U 


vO ON in N 


OnCNVO ON^O OnN^O OClvO OPivo 0>C1^0 ON^ 0^ fi^O o\ 


g; S'^ g^ ::^ 


mmmmmmmmmmmmmmmmmmmmmmmmm 




N Ti-^ 00 


N ^^CO N ThvOOO N ^\C CO <N ^vO 00 N M"^ 00 


"« 




N -"l-^ 00 


N ^VO 00 N ^MD 00 CJ 'l-vO 00 -i-vD 00 N Th^O 00 


< 


Ovo"mOvO movo mo^o mo t^mo't^mo 1^% o'ttmo't^mO K^^o" 


SoS^S'^^ 


i-i m invooo M m mvo 00 w m invo co w m invo 00 


MM 


u 

& 


ro N (N m Thvo 00 H Tj-oo P)vop)oo->i-o>t^>nmcj(NPiNMO) ^i-vo 00 ^ -^ 






a 

o 

Q 


00 oOnOnOO M H M ci M mmm^-<i-inio m^o vo t^ t-^ t^co 00 on Oi 


OOOOOMMMMWHMHHHHHHHMMWWHHWWMMrtH 


i 


"on roMD m\o 


1 

t^O ^r^o mt^o mvo av(N^Do^!-^vDC3^Nvo~-^^^c^f^^ 
o\ mvo r'^^o mvo onn^o onvo a^c^^D on(n\o omvo o> 


s^sls^l 


NNNn(N(NP)(NPlP)(N(NO(NO(N0)NCSNC>)NPlNM 


2 


N -4-vOOO 


N ^^00 N ^\O0O N TJ-^00 P) Tj-^OOO O N ^vO 00 O 



272 



FUNCTIONS OF A ONE-DEGREE CURVE. 





-| 


h 


2 


« ^NOOO ^j ;^NOO0 Cj ^NOCg 0,f^?;^~ ^^VOOO « ;J;nO00 




< 






u 


■^ lOvo NO t^oo ONONO>-<NC<rO-<i- mvo NO t^oo On w M ro •<*■ itjno no t>.00 




m »0 10 10 in 10 10 lANO vOvOvOnOnOnOnOnononOvO t^c^t-^t^t^t^t^r^t^t^^. 




Q 

Q 


■* lANO NO t>.oo 00ON0HP)(Nro-*>A lANO t>«oo ONOOMwro-^iri xovo t-«00 
vn in lAiiAno in in invo NOvONONdNONONONdNOvONO t^t^t^t^t^t^t^r^t^t^t^ 




§ 
5 


§8 S SlcS-a JS^ S^Si.c^'S :^o^|:rS>c?^S,c?^ 5^cg ir=?^cS ?5^,S JT^cg 

■u-ininininininioinininininininininininininininininininininmin 






N -^NO 00 N -^NO 00 N T)-NO 00 P) ^ND 00 PI rhNO CO r> M -^NO 00 

H M M M M N 01 P) PI « rorocororo-<t--<j-rf-^THninu-nn inNO 




•^' 


z 


C ^NOCO O Pj 2-^00 PJ ^NOCO ^^vOC« O « ^NOOO « ;5;nOC«^ 




Q 


O^NO roo f:;^'-' t^l5'^'l215J2 S;^ ^cS ^ "*oo °2. '^ " ^ ^ " "^^ ro r-. 
Q H ro inNO 00 d H ro iono 00 m ro invd oo d h ro inNO 00 h ro inNd oo 6 

g 8 g g 8 g s f5 s s s 5 s s ?! s s s ?rfr?r??frf?^^s-5-^s-S' 




X 


HnSnI-kIcX^s^S^Hs g^l^H^NsI HH^s S-S^H§ 




roro 




Q 
Q 


?HS ?:~^r^r?^ 1% r^~: ^H^HssH^s.^^ 








i 




s;?rN8 s^ s^Sn^ ^%.^%'^t%^t\?,%.^^^^'^^^ "s^^ J5 ;?;§§ 




§ 111 5! 5- ^ ^ q. 1 ^'S ^ 5 4: 1 5^ ^-%^% %% %%%■%. %% % 




z 

S 


« ^NOCO cj ;JNOOO PJ^NOOO . ^NCC« JJ.5:vDC0 P. 5;vO00 





FUNCTIONS OF A ONE-DEGREE CURVE. 



273 





V* 




N ■♦^O 00 O P) -J-vO 00 PI 


^^'S ?>?J.^^<^5-^5^°5-S>S>S^'S>v8 


Q 


T^ w^ o^-o r<-) 1- 00 in ro (^ lo 

O PI m m t^oo P) m in t^oo 
in in in in in in\o vo ^ ^o 'O ^ 
rocncommcnmrnromrocn 


P) m m t~«oo N m in f^oo d N m in t^ Oi O 




M w P) p) fnmm^-<j-in invo 
r^oo o^ M PI m Tf in\0 t^oo 


t^y- ON O .- N rn ^ mo oo ov m p» m mo oo o> 
On M m T^ m\D r^oo On m m ■* mo t^oo o> 












d 

Q 


\or^o>o«P)rn-<j- m^ t^oo 


^^ ^|:r'^5;N8 ^cS 53"):^^ P5°5-S;a^^ 


OOOh-mmmm««mm 


MPjnPipjpjpipipiPimrommmmmmro 






1 


S^'^%^^%^7,:;i^s: 


M Tj-i^ONP) inf^O ro moo w rno On w -^ t^ 
moo 1- -"i-oo « rroo « ^j-t^^ ^t^o 'l-f^o -*- 


ONPtvO ONPJvO ONPJvO 0\NvD OnPJvO 0P)\D 0\P4^0 QnPIO 0\PIvD 0\ t» \0 0^ 1 

o^OOOl-"•-■p^p^M^<->m^-)T^TJ--^Lnm in\0 vo ^ c^ r^ t^oo oo oo cy. o On 

1 




P) -^^00 P) -I-VDOO PI 
„ „ M M w D D 


S-^'S ^^^'^'^^%Z%%^^Z'^%S 


I 

1 


z 


O P) -^vOOO O M T^MD0O M 


^^'S ^ ??, ^^"^ 5- ^ ^^°^ 2> S. Ji^-S^vS 


< 


^^^^%^^S:^^S^ 


SoSJ^SNg^P^S-f^^S^K^-^?:^'?^? 


8S ?S'^'?2 " ^i:'^" 


PI rn in t-^oo PI m m r>.oo o pi m in t>.oo O 
rncnrnrnrncnrorocncnrnfotTic^CTjcnrof^rn 




MMMMMMMMMMI-I^O 

vo in ■* rn P) M ooo t-xvo in 
ooo\Oi-P)m-*Tf u-ivo r^oo 


mOwOmOmONNPI 


ON o M PI rr, -:^ mo r~-oo On - PI m ^ mo tx 


t^ t^03 oooooooooooooooooo 


00 OnOnOnOnOnOnOnOnOnOnO O O O O 


o 

d 


ooonQ h. n mm-* mvo t-oo 


ON moo PI t^i-o o m^Tj-m 

mmPI PI M M O O OnOnOnOnOnOnOnOnOvOnOn 

ON M w m 'i- mo o f».oo on o « pi m -^ mo 


tv t^oo oooooooooooooooooo 


00 OnOnOnOnOnOnOnOnOnOnOnO O O O O 


Q 

s 


1 


i 


z" 


« -"l-vO 00 O (N -*vO 00 Pi 


^^~ PJ^^O<» ^^NOOO O p. JJNOOO^ 





274 



FUNCTIONS OF A ONE-DEGREE CURVE. 



N T^^o 00 o 



Tf-^-^Tt-^Tj-^-^Tj- 



t^ t-~ Ir^oo c 



100 oo>o>oooo<o o o o o o 



M f) N (N N 0) 



. f^OO 00 OO ( 



O^O^a^O^O^C^O^O O O O O O O 



0) N N (N (N N 



(N li-iOO Pi u^OO 



O M "^^ OO O 



■*VD 00 O N ■<*-'0 00 



t^ lo (N o^vo rj-Moo vnno t^mM CT^f^mp) ooovO ^h or^m 
r^fOO t^roO r^-<l-0 r^-^M is.-*m t-^-^HOO towoo inooo incM 



O O O O O 



LO t-^ O 

■^ ■*• Tt ^ 



rO'^-^'^'^'^'^Tt'"^ioi/^mio» 



u-)vO vOvOvOvOvOvOvO t**lxt*^t«*t^t^ 



-*■ ■<!- -.j- IT) IT) in 1 



I invO vO vO vO vO vO VO vD 



. t~. rv t^ tx t^ 

































t^O 


OJ 









rovo 0> ro^O 


a. 


C-) 




















o 


a\ cj 


vn 


0> N ^ 


O M VO O 


(N 


lO 


On 


M 


lO 


n\ 


N in CTi N m 


n\ 


<M in Ov N 


in 


ns 


N 


in o 
















rv- 




-^ 


-rf 












C^ ON OS 


L) ■ 








00 


CO OO CO CO 00 CO 


00 








730OOO0O0O0O0O0OO0 0O0O0O00O0 00 


■z 


<N 


*sO00 


o 


N '^VDOO 


8 


N 


^^<S 


^ 


N ;*^oo 
m ro m ro -^ 


^ 


?^°5-S. 


. 


^ 


^ 


in^ 


S 























FUNCTIONS OF A ONE-DEGREE CURVE. 



275 





N ■♦^ 00 O M Tf^OOO N 


•*vO 00 N -"i-vO 00 N -^O 00 N -^^O 00 


Q 
< 


T^^l ooo'O ^<N ooovo -^w 


OOO t^lOCOM Ot^lOTfW M On t^O rh rO O 


iH fO invo 00 O w ro u-,vo oo 
ID IT) lo in invo vo \o \o \o ^o r^ 
LoiovoLoioi/iinu-iiou-iinin 


M OO invo CO (N ro u^ r^oo n ro u-i t-^co N 
rv t^ tv. t^ t^CO OOOOOOOOOO OnOnOnOnOnOnO 
uiu-)ir)ij~, LOLOir>u-)inininu-)iniouoio ionO vO 


C/J 


S.S<2'S^?J,°^^cS^^^ 


r-^MCO -"l-MOO ION ONt-^'l-N OnvO ■* M On-O 
Tfo OO o. M ro T^^o 00 OA M ro inso oo o w ro lo 


x' 


^^■^^ !;rfrs-?r?r^^=S 


OOOOOOCO 0^0^0^0^0^0^0 >- m >-. n 
<N!NN(NN(NCNN(N<NrOrorororororororo 



Tt- t^ O ro f^ O rovo On N \0 On M moo m looo M Tj- t^ o -<j- tv o ro^ 0\ roo On 
DO H looo M >ooo M Tj-oo w -a-oo H Tj-oo M ^00 m -*oo w rj-oo m ^ r~, t- -s- r-, 

ONOOOi-iHMONNrOrOro-^i-^-^ioin lonO nO no c^ i^ t^oO 00 oo On On On 

OHHWI-IWHMl-l,-.MWHMHWMMMHMMMMMMM«e-WM 

O <N -"l-NO 00 O N -^nD 00 O N ^O oo O N -* nO 03 O IN tJ-^o 00 O 0) ^^O CO O 
t-t w (H w t-t 0) c^ 01 (N CN) rorOiOroro-^-^-^-^-^LOLOLOio lonO 



Pi OOOnO Tj-f) OOOnO ■<*•?) OOOvO loroi 
roovo roo r>.-*0 r^-^woo -^moo ini 



OnnO ro O O ro O r^ 



H ro -"J-nO 00 On i 

O O O O O O ' 



'l-NO 00 ON M 



w ro -^nO oo on m ro 10\0 00 I 
rorororororo■!)-Tf■T^Tl-T^l 
lOioioioiotomiOLoioioi 



rooo rooo rooo rooo rooo 



On -^ O lO O vO 



iOnO vD ^ no 



. 
Q 

OS 

o 


oo CT« O N ro loS £»* 


00 rooo rooo rooo ^ 

ON M (M Tj- lO t-^00 


ON-^Ov-^O lOH t^rooN'i-ONO Noo 
1-1 ro t)-no ooOnm pi -"j-mt^ONO PI ro 




N N 


N (N 


N N CN) 0) 
0) (N O <N 


PI c?c?r?N'c?crN' 


N'N'«'w'N"N'cJ'c?crC?P) C?« P) P) 


Q 


cSh 


^CO 


H ■* t^ O 


"^ t^ O rovo O CO^O 


ON P) NO ON p) moo PI moo m ';^oo m -^ 


O 


Is 


88 





moo PI looo PI moo 

ooooffooo 


M u-)oo w moo M moo h moo m moo 
(?c?o o 0^ o^o" o''o o o 




2 


N 


^vO 


oo N -"I-nO 00 O N ThNO 00 (N M-NO 00 N ^vD 00 IN ^vO 00 


i 









276 



FUNCTIONS OF A ONE-DEGREE CURVE. 





ON rvO 00 N Tt-O 00 C* '^vO 00 N ^MD 00 (N rf^ 00 O N .*vO 00 O 
M M w H H (N N O (N N romrororOTj-Ti-.^.^Tj-iDioiOin mvO 


d 






t^vO \Oinin-.i-mroc4CMHOOOOOOONOOOOOOwMwNNroro 
M -^VD OO N "i-vo 00 M .i*-^ 00 m rOiDr~.OM T^vo 00 N ."1-vO 00 O 


x' 


r-~ t^ r^ t^ t^oo oooooooooNOo>ooooooooi-'wHMCMNNfiwfO 
mronror^roromroro^oro^ororOTt•■^■^•.t■■^■.^■^■^•<l•.l^TJ-T^■^•<:^•^^■^ 


d 

o 




ro (N M ON Ovoo t-^>£) u^u^u^u^u^^Tl-.*rorO(NHHlHHHWMHHHH 
CO N -*u-)t-^a>H roint^ONH romt^ONH romc^ONH mu-)^^o^M foiot^ 

vd t^ t^ t^ t^ r^ t^OO 00O000COONaNt>ONON6660dHHH.^H(Nc!(N(N 

c^rnrocnrOrorororOrO(^rorororororOTt-->i--^-^-<^-^-^-^-.^-<j--.*-Ti-Tl--<J- 



fOO ON N >O0O N lOOO 



nvO On rONO OnC<J "^00 
rONO 



r<lNO O ro\0 O rovo 



r-~ O ro r^ O mNO O rovo _ . _ _ _ _ _ _ 

N romro^Tj-Tt-ioio mNO no vo t-^ i>. t^oo OO oo on On On 
rommrot^rnrornrorororomi^rorororororomrororot^rnroro 



O M ^nO 00 O 



O N Tj-o 00 O 



O O O C 
vO NO NO vD V 



I O (N -^mi-^ONO N -"j-ini-^ONO (N Tt-Lnr^ONM n -i-vo t^ on m n 
1 M M M M ii M « N o o N PJ ror<ii-<-)rororOTl--^-iJ-^'!i--*mm 
'nOnOnONOnOnOnOnOnONOnOnOnOnOnOnOnOnOnOnOnOvO vO nO nO nO 



NO ro w ON r^ 
U-) t^ 0> O (M 



CJ 01 OJ rororor^m'.*-'-*"'-*-T}"--i-ioiOi/^iOio ionO nO no nO nO f^ 



m ro CO ro CO m I 



<N N CNl N N N r')rorororo-'i-Tt-Ti--.j-Tj--^ 



d 


ON<N 


inoo M 


inoo w .^ t>. ro t^ r'lNO Ov N moo N inoo H.^tvO-*t^Oro 


o 

X 

U 


On 


N D N 


■<j-r>.M ■<^r>.M -nj-t^M ^t^o -<j-t->0 Tj-t^o -.j-t^o ■^^l->.o ■«^t^ 
M M CN| IN C) r»imrO-<J--^-l-u-)lA inNO no no t^ t^ t^OO OO OO On On On 




z 

S 


N 


.<»-NO00 


O N .<(-NO 00 O D r^NO 00 N -*NO 00 M "^-NO 00 PI .*NO 00 o 
M „ „ M M c< n N 0) <N mfocorficOT»-.^-*-T«--*iDioioio in\0 



FUNCTIONS OF A ONE-DEGREE CURVE. 



277 



1-1 




P) -^so 00 O P) ^vO 00 W -^^O 00 IN -^VO 00 N tJ-^O 00 O N •^^i OO 
M y' ►, M w (N P) P) P) PI rororororO-*-*.*-*--.l-iAiomiO lO^O 


< 


lOlOThrfromr^pi pj w h h Q O^O^O^O^a^O^C^C^C^C^O^O^O^O^O^O^C^ 
roo t.^-*-H-oo inp) ovO COO t-^-^0 t^T)-Moo lOP) OO mo t^^MOO iDpi 


.^vO f.» O w P) .*vO t^ O M m Tj-O 00 O I- r<-) TfvO 00 ON m ro irjvo 00 w rO lO 
m m ui lOO MDvOvovOO i^i^i^r^t^ t^oo oooooooooo OOOOnO^O O 
t>.t-.t.^c^t^r^r^r^t>..t.~t-^t.^t^r^t^t.~t-^f^t^r^t^c^t^t^t.%t^ t^oo oo oo oo 




.^vo 0> M ^vO 00 y mo 00 rOO O P) moo m M-t-~0 Tht^H TCb-M rt-OO M 
T)-o 00 M ro m t^ N ^vO ON M ro moo pi m t>- on p) .*o on w roo oo ro 

OnonOnO 6 6 C) M M H H M (N (N pj pj rororororo^T5-4-4-mmin mo o 
Tj--*.<l-mmmmmmmmmmmmmmmmmmmmmmmmmmmm 


Q 
as 

o 

Q 


pj T^o 00 PI -<^NO 00 O n ^i^Onn rj-i-^ovp^ ^t-^ovP) moo « 't- t^ o roo 1 
O M -^O ON M ro m r^ o P4 -9-0 00 w ro m t->. o Pi -^^O On m roo oo ro m t^ j 

^ONONONONO^OOO^J^J^I^^^wcj^C^C^C^rorom 

1 

1 


o 

X 


ON P) moo M Th r^ O roo on pi moo w r*- t-. o roo On P4 moo w Th r^ o roo On 


mONP) mONPi mONPo moo p) moo w m>oo P) irioo w m,oo m moo w moo w ■* 
OnOnO O " ^- " P» P^ P< rororOTl-^T(-mm mvO O O f- t^ t^oo oo oo On On 





i s 



i° 

! < 



o o>oo t.^ t.^0 



t^oo O PI Tt-mr^ONO P^ T^lrlt^ON0 Pi -i- 
(N PI roforofororo.^T^T^.^.^.^mmm 



Tj-O t^oo O 
m t^ On " Thv 

, m 'ir^'^ o o o 



T)-0 00 O P) .* 
ro m r^ O pi tj- 

00 00 On ON On 






S ! 






lOO o o o o 



■^O p.- On O P) 



t-~ t.^00 OO 00 00 ON 



Q 


o 


On 


N moo 


M 


moo H ^ 


t-.Q 


roo 


On 


P) 


O 


ON PI m 


00 


M 


■>*• 


r-O 


ro t>~ roo 


ON 


o 


o 


On roo 


ON roo 


ON roo 


On roo On 


PI 


O 


r-h 


N O ON 


N 


vn 


rTN 


(N O 


ON N 


O On PI 


in 


ON ON o 




















m m mo O O 






30 00 ON ON 










































U 










































z 


o 













P) 




O 


















§ 





































278 



FUNCTIONS OF A ONE-DEGREE CURVE. 





1 




N ^nO 00 (N ^vo 00 (N Tt-^O 00 (N -.J-^D OO W ^vD 00 P) ^vO 00 
„„„„„CH(NN(NiNroMmrorOTt-Tf^Ti-Tt-inin in 2^ mvo 


< 


in invO \0 vO tv. t-^ i^co oooo OnOnOnQ C h m pj rorh inyD no t-^oo on "-i N 
roo t^n-MOO inw onnO ro o t^ rt- n onno mO t^rf^O) inpj onno ^ m oo in 


NO 00 ON H ro -4-^0 OO ON M rn ino co d w ro m t^oo 6 pi rn m t^oo 6 pi 4- in i-^ 
in in \D'^ NOnC/nOnOnO c-..tv.t^l-^ t^co oococoooco OnOnOnOnOnOnO o 


Id 
C/} 


■^ONinOMO n '^ PI t^ rnoo mONinM i-^pioo ^^j-Qno ro onno pi on-o p) on in pi 
NO CO M T)-vo ON M '^NO O H ^j- ^o Onpi -^t^ONpi mt-^O P< moo O rONO 03 W ^ 


rorOTt-T]-.I^T)-lnlnln in^o no no no t^ t^ t-^ t-^co ooooonO-OnOnOOOOmm 
nOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnO t^t^f-t^t-^r^ 

I 


O 
Q 


1 

Tl-oo rooo rooo PI t^ PI t^ P) t-^ roco TfON^O u-nhno m t^roONinONO PJOO -^ 
c^ H .^vo ON H ^No ON H T^NO On m ^no Oncj Tf-t-^ONp) T^t^ONP^ inc--o N in 


N mrororo-<j--*-*'*-ininin inNO no no no r^ r^ t^ t^oo oooooo onononO 6 

vOnOnOnOnOnOnOnOnOnOnOnONOnOnOnOnONOnOnOnOnOnOvOnO^DNDnO t^t^t^ i 




i 
1 
On pi inco M -^ t-^ On N moo H ^ (^ rONO On P) inoo H rONO ON p) inoo M -+ t-^ 


I^nS^ 1 r! ^ ^ H!U ^ ^1 K K K 1 ^1 ^ S K K^tl! ^1: ^l-l.€ 1 


MM 


2 


i 

C^ -*-NO 00 P) ^NO 00 N 'i-NO CO M ^NO 00 PI ^nO 00 PI tCnO OD j 






i 

p. ^NOOO PJ ^NOOO PJ ^NOCO ^!, ^^~ ^ ^NO CO Pi^ ;^NO c« ^ 


Q 


i 


inNO 00 N ro in f^co o p) romi-^ONO p) .*-ini--ONM pi .^no r^ cjn m p) T^NO 
ooooooc3cScocococ»oocoooooooooco ra^oo co co'c^oo moocooo co'oo'oo'oo'co 


CO 


I 

M ^00 CJ NO 1-O0 PI NO -+ ON rooo MNO M inO T^-ONn-ONT)-ON-+ONT^ONT(- 

rOint~^0 M int^ONPI .^P^Onw r^NO On m ^nO On m rONO CO m rONO 00 m rONO 

NO NO NO t^ r^ t-^ t~^ r-^co oooooooNONONONOOOOMMMwpipjpjNrororo 
ininininininLninininininininin uono nonOnOnOnOnonononononononono j 


2 

as 

o 

d 
d 

K 
O 
X 

u 


NO CTn rONO ro t^ ■.»- t->. M T)-cO P) no •^CO PJ no rt- On rOOD P4 NO m, in T^ 1 
t^ ON PI ^ t-. On M Tf-'O CO M ro inco o roini~,o Pi int^ONPi T^^^ONPl .*-r-,ON 


in inNO NO NO NO t^ t^ t^ t^od oococooNONONONoddoOMMMMpiNcJpi i 
inmininininininininininininininin inNO nononononononononononono 


1 

On PJ inoo I-; 'f f;- rONO On pj inCO ►^ -f t;- rONO ON P) inoo ►^ ^ t;~ rONO On 1 
S) S)NO NO^NONONONDNONO vcTno^nO^nonOnO NO NO NO NO NO S no'nO^noS S S no'no' 


„HMMMMMMHMHMMMMM„MMMMHMMMMMHMMM j 

1 




C< -^nO CO N n-NO OO PJ -^nO 00 M .^hNO OO O PJ -+nO OO O PJ -shNO 00 





I 



FUNCTIONS OF A ONE-DEGREE CURVE. 



279 





1H 


Z 
S 


« rhvOOO ^. 2-^00 O Cj ^vocg O JJ^^vOC^ O ^^vOOO O C^;J;vOeO O 


Q 


»0 t^ O N r^invooo o m n ^f u-|^ OO N Th\0 00 P4 ■* 

00 lOroO f^■<^woo uiN t^-*-MOO irimO t^M-M Oivo ro oo i/i M 0>vO •'t- 


Hiisil^ §; S; S S; ^ iiii?^l^f S^ § 111 § § §1 1 1 




vo ON S tn S^'o fo^D O N inoo rovo" On (N iot^ r? ? t^ rO'O On N lOOO fO 


t^ t-OO OOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 0&-0QO0 CO 00 00 


Q 
O 

Q 


OONO fOHOOvO Tj-H ONt^iorOH OCOVO ior»lH Ooo t^iDTj-N H ON t^NO in 
lOOO w •♦-O ON N m t-^ rONO On fJ ■* t^ rOvO ON m ■* t>. rONO On H ■>!- t^ O 


0000 OnOnOnOnQ " w >- *~ CN M N nc-omm-*->*--*ioinm rnvo no vo t^ 
t- t^ c^ t^ c^ t^oo oocooooooooooooooooooooooooocooooooooooooooooooo 


d 
o 

5 


moo H fONO On N moo ponO on O in t^ cOnO On 0) -^ t^ rONO On m ij- t^ 


l^cl^l^ 8n n Sn s^ s; §N 1 s; s s s H 1 1 S sHi §^ ^HI^In 1 




2 

^ 


O c* ^NOOO « ;^NDoo c; ^ND^ %^'^'%%%%T%'%^^^%%^ 


00 

1 

1 
1 




N ^NO 00 O N ^NO 00 O N ThNO oo N ^VO 00 N t(-nO 00 N 'l-NO 00 O 


< 


^^Z^^^'^^^ Snn^S^^^^^cS^nS ?J>??:^^g§ sas ?l^I?S 


0NON0N0NON0>0N0NaN0N0N0N0N0N0N0N0NO>ON0NON0N0N0NON0NON0N0NO>0N 


J 
a 


N ONNO roo r-^MOo inN ONt^^w ONt^-*f) ON^^T^<N ooono c^m ONt^in 
■*nD On (n in t^ m moo h roNO on o -^ t^ O ro moo h ^ t^ on M moo cono 




Q 

o 

Q 


-*0nO roONmtMOO '♦■H t^roo t^-*-w f-^fMOO mo On^.'S-h o<nO en h oo 
moo ro moo " mno onm -^c^onm mr^O ro moo m cOnd On d ti- r^, q f) ir> 


OOi-ii-iMi-ip)NNpqrom(T)m-<^Tf--*mmm mvo no no vo t^ t^ t>.00 00 oo 


Q 


r^ O fovo ON N m (^ fovo on n moo h ■* t>. o mvo On h -^ t>. o fO\o o ci in | 


r£gi»f^^i'ii'i:"3i^tKHsi„-i=Hs'ii' 






N -l-NO 00 N ■♦no 00 M tJ-vO 00 N t)-vO 00 O N •♦nO 00 N •♦nO OO O 

„ „ „ „ M <N w <N (N o mroMroro-^^Tj--^Tfmmmm mNO 





280 



FUNCTIONS OF A ONE-DEGREE CURVE. 



P) -"t-MD 00 O 0) -^VO 00 O (N 



O t^ ■* o 0>^ -"J-MOO iDMO (>.■*« OnvO r<lHOO inroo (^-^M 0>0 ro h oo 

M ro in f^oo O N ^iot^O\w N Thvo t^ o H ro t)-^o oo O h ro lovo oo O N ro 
vOvDvO^^ t^t^t-^r^t^ r^oo ooooooooco C!\ a^ o^ C!\ <j\ o O O O O O h m m 
OOOOOOOOOOOOOOOOOOOOOOwMMHHHwwM 

00 O w ro ThvD 00 O 
lO O OJ looo iH -^t^M 'T^^^-^ Tj-t^O rot^O rovo O cOvO On r^vO On m\0 0> W 

t^ t^oo coco onOvOnQ O M H M IN N M rororo-*-*-^-*ir)io invo ^D vo t^ 
O^O^O^O^q^O^O^O^OOOOOOOOOOOOOOOOOOOOOOO 

UTO >0 t^ t^oo OnOnOOOOOO 

On <N lOOO M -.^ t^ O Tt t^ O mvo 0< (N inoo H Th t>. H ■* t^ O rr)\0 ON P) lOOO N 

lONO NO NO t^ t-^ t^oo 0000 OnOnOnOnO O O h h h (M P) P) roror<-)(T),j-Tj.^ir) 
OnOnOnOnOnOnOnOnOnOnOnOnOnOnOOOOOOOOOOOOOOOOO 

lOCO O rONO ON H T»- t^ O PI inoo O p^nO on h tI- t^ o n lOOO h ponO On h ■* t^ 

00 M irioo M -^co M -^00 1-1 -^t-^H -<i-t^o -^-t^O ■<*-t-~0 ror^O pond O pono 
00 OnOnOnO O O w w m pi « P) roporo-^-*-.^ir)m iono no no f^ t-- r^oo oo oo 

0000"MMMMMMMHWMHHMH1-IMHHHMHMMHMM 

P)Plp)P)PaPlPiPlMP)PlPlP)PlPlPIPlPIWPlMMdNP)PlNWPIPlP) 



O <M -^NO 00 O 



TfNO OO O PI -^NO ( 



O PI ■'l-NO CO O M -*nO < 



coropofOPO■^■<^-■*■*■<^mlOlO>o mv 





O P) ■'(-NO 00 P) ^NO 00 P) T)-NO CO N -^ND OO N -4-NO CO PI ""(-NO 00 


< 


^MCO inPi t^'J-HOONO roc r-.Tl-H onno poOoo iapj o^c^'I-hoono roO 


O N rOint^ONO N ^int^ONM pg T^ND t^ on « ro 'S-nO oo On m M iriNO 00 P) 

o ?o*?o'o'o'o'o'o"o"o'o'o'o^o^o^ o"o "o 



OnOnOnOnOnOnOnOnOnOnO i 
pono On w inoo w -!^ t^ o T)- 

OOOOOO OnOnOnO O O 



1 irjNO NO NO NO 



o onoo t~~NO NO in 1 



O POnO On POnO 
On On On On Q Q 
On On On On O D 



lOOO H -^ t^ O rONO ON I 



rONO ONP4NO ONPINO ONPI inoNi 



OOOOOOOOO 



rn m CT) -^ '^ -^ irt \ 



t^ O rovo On 



inoo H moo 
tv r^oo 00 00 

o o o o o 



FUNCTIONS OF A ONE-DEGREE CURVE. 



281 



■<l-'0 00 O P) -<4-^ I 



O N -^-^ CO O N -^vO 00 O N 






I 00 in n o 00 tn i 
■4- m i' 



O t^ m p) o 



o o o o o 



b» H -^OO H moo N in o^ 

t^oo odoo onct>ct>6 o o 



woo rovo O m ["> I 
M M M^ pi rnror^- 



inoo N vo o mvo O 
in invo vo vo t-. t^oo 



in m mvo vo vo r^ i 



• t^ H -^ r^ H -"j-oo H -*oo H inoo 
. t^oo OOOOO^O^O^OOOwHM 



inoo N moo n m C3^ w 

(N N romrOTj-Tj-Tt-m 



•"1-vO Ovw r}-t^o>p) Tj-t^o 



O m moo O m^o i 



(-00 H ■<t-r-,H M-t^o -"i-i 
iooo\o>a>000ww 



r^ O tn t^ O rOvO O rOMD on fOvO O N ^D Ov I 



O M -*vO00 O 



O N -^^ 00 O <N 



< 


CO m 


roO t^-*- 


<M OO ^ 


Moovo cT) t^mp) ot^-*w ono -* w 00 no m CO 


j?;i'^2^as 


^^frs^ 


M p) Tj-vo 00 O M r<-) Ti-o 00 M fo m r^oo P) ■>*- m 
rocom(r)roro^^-^T^T^mmmmm mNO no no no 




i 


N O 


ON NvO ON 


NVO 0<N 


NO O Pi NO O mNO O rONO O pono ror^O fOt^O ■* 


en 


s^s-^^^^ 


S'o 2 


OOHMHNp)p)rommT)-Ti-mm mNO no no t^ t^ 


^ 








Q 
X 

O 

Q 


NVD 


ON moo 


M ■* r^ O 


■*t^O -"j-t-o -*t~^0 -^t^O cnt-O mt^o mr-0 


^^ 


o'^^^ 


ooo'o'o'oo'o'o'SSSmmm^^s^^^S';^^"::^ 




Q 


t-.0 


N moo 


r^NO Ow 


^ t^ O PI m p~. rONO 00 M -<hNO OP) ■* c^ m moo 


i 


CO CT. S^ O^ ? 




ON mOP) mop) moo P) moo w moo m moo w -^ 
H D p< N mfom^Tj-Tj-mm mNO no no t-« r^ r^oo oo 

SnnSnSSSpipinpISSSnnnnpjpi 


« N 


N CJ t) (N 


a 

S 


N 


TfvOOO O 


N ■<*-\O00 


g S S-^'S %%^'^%^%t%%S.^t'^'^S 



282 FUNCTIONS OF A ONE-DEGREE CURVE. 



f^OOO loroooo mrowoovo mnoovD mwoovo nMOOvo -"i-H o>vO ■* w CTi 
p tM p IT) t-^ C^O IN _-?-vO _t>j5N M ro -3-^ 00 O H rfi to t>CO O N Tj-irjt^osM CJ 



t^ r^ r^ f^ f^ t^oo < 



in OS rovO -^00 IN ir)0\rof-.M ino r<-)\o O -"l-OO N \0 O -"t-OO M in O (^ I^ 



N inosNvo O mt^o M-co H inoo N \o o m t^ w inoo n in ct> m t>. o -^oo 



-^\0 O w 'JfO t> M -*^ OS H TfvO OS HI -^^O OS H TfsO OS M "^vO OS w -^vO OS 

O roso O cn\0 OS roso osoiso osN inos(N inoo C) inoo w moo m T^oo « ■■^ r~ 
oqoqoo qsc^ososqoqwMM_nNNrornrn-!j-^^inin mso so so r^ t^ i^ 



O (N -*sO 00 O M ^sO 00 O IN -^SO 00 O (N ^sO 00 O P) -^sO 00 O (N -^sO 00 O 

1-1 I-. M M H p) 0) 0) 0) c) romrotr)m->f-^-<j-Tt--<j-inininin mso 



CN M IN mcnmmro- 



t^-*N osl-^^N Osso 'I- H Osso -^MOOsD rOMOOso roMOO mroGoo mm | 

00 OS M rn 4-so 00 6 M m in t^oo 6 oi ^mr^osM n -4-so co Os m m insd oo 6 I I 
M H d cs) N N M roromr')rorO'*--<)-^Ti-Tj-T^mininmm u-ikO vO so so vO t^ 

NPiMNNCMNONCNNONDNOJCSlCslMCItsiNNNNDMCllNPlN 






Th r~- 


M moo N 


so 


OS m 


r^ 


d 'l-oo 


M m 


OS NSO 





Tj- t^M m OS 


IN so 





■* t^ 


M 


CO 00 00 


OS OS Os 


ro m ro 


rn 


N IN N ro ro 


tn m fo 


m in m\o so so 
m rrt rr> rr, en <^ 


(^ l-^CO 00 00 

m n ro m m 


? 



IN 


n OS 0) SO OS rnso ro t-^ -*• i^ h -*oo m moo n so Os roso ■* r^ m -^oo 


JJ" 


?J?^^^ ?r?:r'^^'S ^^^^^^r'n^^^?-.?r.::;:^-^^^\^^]^ 



so OS M -^so osp) Tj-t^os(s) mt^o o moo O m moo >-. mso oo m -^yO Os w 

mco N moo m moo m moo i- 'i-oo l-■<^-^>.l-lT^t^OT^r^oro^~.0 
----- -. .— . . ^ .- ._ .-.^ vo so I^ (^ t^OO I 

N W W <M N N N 



OOOOOO OsONOsO O O M " 



N en rn m -^ •4- •*■ \ 



I 



FUNCTIOIfS OF A ONE-DEGREE CURVE. 



283 





^ ^VOCO O CJ 2-^00 Jj ^vO<« O ^;?;^~ O ^5^00 O C^;J;vOC«^ 


< 


vO •*•« CM^iON 000 iDfOOOOvO r^in 0\0 -^N ONt^iorOOOOvO row' o^\0 


K ^ ^cg eg <|;s g eg; 5; g s|j|> 8 s ? ^ 1: 1 2_ 2 2:^'2 2; s s: s:^<| 



lO 0< rf> t^ M lo O -^00 N vO w lo o^ r<ioo Pi vO O iri o <^ t^ i 
■ m in\o 'O r^ t^ t^oo oo o^ o o^ O < 



vO VO vO ^ vO vO vO vO vO vO vO vO vO vO vO vO vO ' 



fOvO •*00 N vO O •*00 
00 00 O O O O 



VO q '^■oo N\q q -*oo wvq 

ro tT) CO ■<l- ■^ m m lovo ' 



' \0 vO vO vO vO vO vO vO vO vO vO vO \0 vo vO vO vO vO vO vO 



■<t-00 N VO O -^OO N 

t>.00 00 Ov Ov Ov 6 

" VO vO vO t^ 



fO lOOO O NUItxO N -^t^OVH ThvO 00 



vO vO vO vO VO vO vO vO 
CJ « C» M (M - - - 



t^ O f*l t^ O f^vo O rnvo Ov rovo Ov P) m o N 
HiM M N rororo-*-*-*-*ir)>/^ uivo vo vo t^ 

P4MNP1<NPINNMW<MNNWWNNW 



O N ■*vO 00 O P) -^VO 00 O N -^VO 00 O N -^VO 00 O P) -^vO 00 O P) tJ-vO 00 O 
M w M M M c< PI w o PI rororomr'i-*-<*--<j-'^T)-ir)ir)mio mvo 

OvvO ■* H OvvO •+M Ovt^ThP) Ovt-«-<J-Pl O t^ir>P) 000 vnrOMOOvO -"i-i-i ovvO 

C< -^VO 00 ov w r^ lOvO 00 O N COlOt-Ovw P) -*vO 00 Ov M m invO 00 O P< (T) IT) 
P) P* P* w Pfl roror^rOrO'^'^'^'^'^'^ioioiriiou-) lovO vOvOvOvO t«*t^t^t^ 

t^w iriOvfOt^H lOOvrot^M inovrot^M inovrDt^w loO -^oo P) vo O -^oo 

O w M w N P< fOrOfO-^-^-lOin lOVO VO t>. t^ t^OO OOOvOvOOOl->l-lP)P)P) 
inioioir)ir)ir>tnioiomir)ioui>ninir)ir-, iniommir) invo vo vo vo vO vo vo vo 

00 M ui ov ro t^ M -^oo Nvo ■<l-r>.H irjOvfOt^O -^00 wvo O ro(^H ino<ro 
vo t^ t>» t^oo ooovovovo o M M H p) PI p) fOfo-4-•<^•^»n »/ivo vo vo t^ t^ t^oo 

Ov w M-vO OV H -^vO Ov H ThvD Ov m -^vO OO w mvO 00 H rOvO 00 hi mvO CO H fO 
t^" -^t^O -"J-t^O rot^O rovO O rovD Ov rovO Ovp»vo OvP) mOvN moo P) lO 
t^OO OOOO OVOvOvO O O >~> >-* i-t 01 P* Pi N rOf^rO-^-^-Tt-mu^ ir)vO vO vO t^ t^ 
li^ in in in lO lO invo vOvOvOvOvOvOvOvOvOvOvOvOvOvOvOVOvOvOvOVOvOvO^O 
PlP)PIP)P)P)P)P)WP)P)P)PlP)PIP)P)PlPl«PIP)P)PfP*HP)P'P'P»P< 



O PI -^vO 00 O P< -^vO < 



O P) -*-vO 00 O P> ■♦vo < 



284 



jruNCTIONS OF A ONE-DEGRKE CURVE. 



2; 


.. ^^00 c; ;^vO00 JJ ^^co p^^vooo ^^vooo « ;j;.0<»^ 


Q 
■< 


O^t^'fN ooovo rOH ot^mf) O00\O -"J-m a.r-«>^r^000vO ^^N O t>.inm 


n m in t^ cji tN T^vc) t^ m n xr^'^ 00 N romt^cJiM n -^vo 00 m ro in 
000000000000>CT>0>ONONOOOOOi-iwi-i)-iMi-ip|ONNMrOrorom 

■*'»•■<l•Tl-Tl-rJ-•.J-T<-^,^,t-lnlnlnlnlnlnlnlnmlnlnlnln■nlnlnlnlnlnln 





m O ■* O rooo N t^ M in O "1 o> ■»^oo rot^O t^MvO O ino -^oo rooo M 



00 0^ O 0^ O O 



ro m m ■-*• -^ 



mvD ^O tv fxoo 00 00 000 O 



0^0^0^0^0^0^0^0^0^0^0^^0^0^0^0^0^0^0^0^0^0^0 O O O O 



I \0 O -^OO ro f^ 



0> •>!- O ■*00 N VD O -^OO N 
M M f) eJ rn(r>TfTi-Ti-in 



ro IT) t^ O N -^vO 00 M m in t^ o n ^vd o i 



• 00 0^0^CT^O^0^0^O^0^O^O^O^O^O^O^O^0^O^O^O*O>0^ 



-I 



■* ■<*■ ■>!■ -^ in I 



O N -^^ 00 O 



N -^^ 00 O N -^^ 00 O 



00 O w -^int^ONi-i 01 Th^o 00 0> H ( 






I N O 00 \0 ro 1 
I in P-.00 



'^oo mt^Hvo ^ovror^NMD O ino\ moo n vo m in 


-<1- rOOD M vO H in 


m invo ^ t^ r^oo oooooCTiOOMMMc>)Nmrn-*-^m 
t^t^t^r-st^t^t^t^t^r^ r-.oo 0000000000000000000000 


in in\0 vD f^ t^oo 00 
oooooooocooooooo 



N\0 -fOO 

d 0" M M M 







"^-OO 


ro 


"^ 


tC 


ir, o> CO t-- 
tQ tQ r^ r^ 




in o> ^co 

t^ t-~oo 00 
t^ r^ r^ r^ 


N ^.O ThOO M VO m 


M-^O 


9 


M 


mvooo 





ro 


in t^ 


N 


f "^ T 'I 


Tj-vo 00 M fo inoo 


N in t~«ON N 


■^ 



t~~ f-~ r^oo 00 00 



O rO^O O^ CM vO O 



M <N W <M 



)000000000000000000000000000000 



30 00 00 00 00 00 
PI (S M W M (N 



FUNCTIONS OF A ONE-DEGREE CURVE. 



283 



O N ■<*-vO 00 O P) 'l-vO I 



< 


QOOVO ^P4 OOOtfJ.it-P) OOO^ '^Pl OOOO "^P) OOOVO u-irOM OM^lOfOM 


ON P) -^vo oo o M ro lo e^oo O P» "^vo t-^ o m f^ u-^vo oo O P) -^mt^OM ro 




u" 

CO 


rooo P) t^P) t^NvO -^ wvo M^O O inO inO >nO "^0 "lO tJ-o\-*0>-*On 


\o >o t> t^oo ooONO>oop4i-'>-<p)rnro-*"*-m i'^^ 'O t^ i^co 00 oo on O" o 

p!N5p!Nr»NNNP)<NPiNP)PiNNP)PJP4P)NNC<P)P)P)PlSprpr 



n 




o 


-*00 rot^pivo w inO -^ON rooo Pi t^ w vO >n O^ 'S-OO rot-.PJvO m lOO inO 


coooo^O^OO>-'l-'P'P^P^ror^^■'l-■^^l^) invo no vO r^ t^oo ooonOnOO'-'mpi 


Q 


P)N§8p3S5NPINP)PiPlP)P)NP)pi<NpiP)PJPlP)PlPlPlSp)p!S 


^ 





Q 


o 00 P) -^nd CO fo>oi>.ONM nmt-.ONM mint^ONw -^no oo o p) -"J-no oo 


O 

s 


pj in ON P) u-ioo >- inoo " Tt-t^H -*t^0 rot^O rONO On roNO On P4 no 0> P) u^oo 
NO NO vo t^ r- t^oo oooo OnOnOnO h h m p) p) n m <r>^or<^^-<^-*u-|u^lr) 


^^^^^^^^^^r<imnrororocororo(nrortroMc<icororofor?)fo 


2 


o P. *Nooo pj 2-^00 Pj ^NOc» cj^^Noco ^^Nooo 2- ^NO« o 



row ONt^mroOoONO -^N OOOnO -^P) O t^uirOM ONt^irimi- ONt^inPi O 
in t^oo d N -^no t^ ON m ro inNO oo O P) Tfint^ONM n -^vd oo o w rn in t^ on 
mmm■<t••^■^^■*T^■*^nvnlnln rnvo novononono c^t^t^t^ t^oo oo oo oo oo oo 
inininininininiO'ininininininininininininLninininintnininLninin 



i 

in 
•>< 


WNO w ino inON-<l-ON fOOO 


rO t^N ts.HVO H 


in o >n -"J- ON 


■>*■ On -^OO rOOO m 


iggggHssll 


ri-ii?|2 


H M PJ N 2 

P) M N N PI N 


ro ro 2" ^ in inNO 
P) N P) P) N 5 P) 


Q 

o 

p 


P) NO w m -<*-oo PI «o H »n 


On '^OO m t^ h no 


in ON moo P) 


(^ r- .no ^ ON Tf 


S S^ ON&o!oNO^ONONONO^O ^ O 


sagggg 


gsggssl 







d 




fom t^ P) 


■*NO00 ro 


in 


t^O>H 


fDNOOO 





, 


TJ-NO ONH 


PO 


in 


^ 


ON PI Tl-NO 


p 








H inoo 


H 




1^ 












rh 








8 


8HS 














T 








OnOnOnOnOnOnONOnOnOnO' 




















o 




P) P) N f< P) 


p< d 












c, 




g 


O 


P) tJ-nO 00 O 


PI tJ-nOOO 




-j-NOOO 





PI -4-nOOO 


o 


PI 'l-NOOO 


o 




^NOOO o 


s 








HWC. 


" 








"' 


"' 




^ ^ 'I- 


^ 





286 



FUNCTIONS OF A ONE-DEGRKE CURVE. 



O Pi ^\0 00 O N -^O 00 M ■^'O 00 O P) -^^ 00 O P) Tj-^O 00 O N -^VO 00 O 

H »- H H 1-1 p4 N PI pj pj mromrofO'*-^i-'*--<*-->j-iom>oin mvo 

rOM ONf-iO'J-M 000>O \r, r<^ n 0\ t^vO tJ-n OoomD irirOM o t^vO -^ P) O 00 
(-» O O PI "i-vo 00 1- romt^oO P> -^mD oo O ►^ roir)f^cj\0 Pi -^^O 00 O w 
a>0>OOOOOMHMMwi-ip)NPj(NPirororororor<^-<j-Th^-*rhu-)»o 

HVO H r^P) t^Pl I-^PIOO r^jOO roo-^0-*0 »^0 iomvO M^O PI t^PlOO rooo 
VO 'O f^ t^oo ooOO>00'-ii-iNP)r^. ro->*-iA invo vD t^ r^oo 00 o O O O w w 
P)PlPiP)P)PlPIP)P)P4NP)PiP)PIP)P)P»P)P)P)piP)NNNPjP)P<P)PI 



O "4- 0> -i- On ■* 0^ rooo POOO 
VO VD \0 >^ (-^00 00 O O O 



tv. P) t^ PI VO 



-!i- T^ in lovo vo 



lO O •>*■ 0\ ■* O -^ 
. t^OO 00 00 On 0\ 6 



■^vo 00 O PI "«i-vD i^o»w mlo^^o^H■fo mvo oo o N -^vo 



■<i-oo M ■<*- t^ o -^ t^ O povo 
in IDNO VO VO 1^ t^ t^OO 00 oo 
PlPip)PlP)PiNPiPiP)PiP)MPiP)ror<-i 



CO m CO CO CO < 



H moo M -<*• I 

PI PI PI CO CO I 
CO CO CO CO I 





H ONt^mcOM ONt^iocoH ONt^iocoH o^^^lo•<fp^ ooovo -"^w ooo t-^inco 


CO ■'^no ooOwcoiot^ONHpi ■"^'O 00 o H comt^ONi-i P) -"i-vo oo o h ro m t> 








ON-<i-os-'*-o>'*o»^ONrt-ON-*ONThONTfo mo mo mo mo m«vo h^o m 


OMHP)Ncoco^-*m m^ vO t^ t^oo a- O O m h p) n cocOT^■.*-m m\o 

COCOCOCOCOCOCOfOCOCOrOCOCOCOCOCOCOCOrf•-<j-^^,)-■^^T^-.l--^^T^■<J• 
P>P)P)P)PlP)P)NPlP)'PiP)P)DP4PlPlP)NP)PlPlPlPlP)P)P)(NP)P)P) 



O m 0> -^oo cooo p) 



~ N Pt 

Q PI PI 



M m O •<^ 0> -"i-oo cot^. N t^pivo m mo mO "^0 
mvo ^ t^ t^oo 0000 ooo O M M n p) coco-^-^in m\D 
PJ PJ PI PI P) PI PI PJ PI PI cococococococococococococo 
PlPlP)PlPlP)PIPIPlPlP)PIP)P»INP)PgWP)PIP)P4Pl 



O PJ -"it-NO 00 o p) ■^^vo OO o PI m t^ o> I 

CONO O covO On P) M 



pomt^ONii fomt^ON 



cocococococococoi 



o PI -"i-NO 00 o w -^No 00 o 



O O H M H 

MPIP)P«NP»WMNOP)PlPiPlC4MPlPl 

icorocOfOcorocococorofOcocofofOfOfO 



FUNCTIONS OF A ONE-DEGREE CURVE. 



287 





r 
1 

1 

! 

i 

1 




1 

N ">1"0 00 O M -+vO 00 P) ■^"O 00 O W -*^ CO O N -^-^O OO N -^^vO 00 


< 


1 

t>.ir)rON 000 t^mroN Ooo t^inmn Ooo t^in-^N h o t^vo ^ ro m ooO 


vooo N -^lOt^ONM (r> iri\0 00 o -*0 t^ o >-< m in t>.oo O P) ■^'O oo c>. m 
>- ►- " ►- '1 M M C) CJ c< o rororor^r<^o^■<J--i--^-^^■TJ-mlr)mu^lr) invo 
cooooocooooocooooooooooocooooooooooooooooooooooooooooooooooooo 




u 

II) 


H vo M t>. r<loo -^o^iriOvO N r-.fOO>-^0 mM t^woo -^j-omo^ n i^roo> 


r-< 1^ t^ t>.oo 00 00 oo <S oo'<» 00 00 oo^o? oo do"oo"oo ooooco'o^o^o^o^o^SSa^o^ 

.N(NMN01N<SNMtMP<(NNN(NC<(N(NP)D(Nf)INWM(N(NPlNP)M 


a 

OS 

O 

Q 


N tN.Pi t^p) t>.P) t^p) i-Npj rs.p) t^pi t^p4 t^pi f-.p4 r^P) r^p) i^pj i^pjoo ■* 


MNP»NP4P)P)PlP)P»PiP<p)WPlP)PJ P< N PlP)PIP)p)NO(NMP)P)P) 




5 

s 

CJ 


w ro lAvo 00 O P) romt^oo PI -^\0 t- On m M invo 000Plf<^ln^^o^0P^•<*■ 


vO O p) looo PI inoo M ^ t^ M ^ t- <r)\o rovO o P) vo On P) inoo m inoo w 
•* -.j- ID in invO vO MD t~. t^ t^OO OOOOOiO>OvOOOOMi-'>-iP<PlP)POrOfO'<f 

■<l--<^■<^M-^■>J-T^-*•T^r^-^-<^^,^T^r^TJ-lnlnlnlnlnlnlnlnlnlnlnlnlnu^ 




N -"J-vOOO O P» -<fvO 00 PI T)-NO00 O P) '^vOOO P« -^vOOO O PI -^vOCO 
MMMMMC^(^^P^P^P^p^r<^r^ror<^T^rJ-.^T»-,Cu^lo^Oln mvo 


1 . 
1 

! 
1 


a 


« *vO«, PJ ;^VOOO O P. ^.OCO O Pj^;5;.Oc« O ^^vOOO C^JvOOO^^ 


Q 


00 t^inroiH Ooovo -4- mn o> t^\o ■<^p^ ON(^in-<j-P) o ONt^incor* ooo t^ 


K S S ^ K^l.l!l.l. ^ E: ^ S ^ ^'^'I'd.l. 1 P: ^ S K ^<g eg J"! 




a* 


00 foo>-*0 loOvo M t^N t^ fooo mon-*0 inovo H t^c»oo mo\^ O «nti 


M e) p» fi ■* •* m mvo >o t^ t>.oo oo o>o>0 h « n pi co m -^ -^ tn mvo t^ t^oo 

^p)pjp<pqpipiw^pip)p<^'S^Mprprprwc<c<prprpiprprprprc<pr 


Q 

K 

o 

d 


•*C0 N l^N t^N t^P) t^N t^N t^PJ f^N t^P) l^Pt t<.N l^sN t^Pl t^P) txN 




Q 


vooo o M roint>.ONw Pi "^vo 00 P) foiot^ONM fo -<j-vo 00 o pi --i-int^OM-i 


lo m lovo >£> vo vo t-^ t^ 1^00 oooooio>a>OOOHiHi-iP4NP)rr)fOfnro-<j--^ 


J5 


1 





288 



FUNCTIONS OJ" A ONE-DEGREE CURVE. 



N •*VO 00 O N ■<l^VO 00 N -^^O 00 O N ■*vO OO O N -^VO 00 O N Tj-vO 00 O 
M M H M M M W N N N COfOrOrOfOTt-<t-'«l--^-^ioiOlOlO in\0 

rOH 000 t^iOTj-N M OOO vO m ro N O OOO \0 iotI-N m OOO t^u^'^l-row O 

r~ O w PI -^vD 00 O (N roinr-CJiH ro lovo OO O N -^^ OO c5 h (r)"1t^C>M ro 
I- w (N e) N N N rocnrofOfOm->i-->i--<j--.i--»j-ir)ir)ir)ir) ir)\o vo vD vo vo vo r^ r^ 

rOOO •'I-OVO NOO -^O VOH t>.MOMnH t^MOMONOO ■^0>0 (NOO -^OvO (N 

N N fO T^ ■<!- ID ir>vO t^ r^OO OOO^O^OMM(^^N<^•4■•>i- >nv6 VO t^ t^OO OOO 
WMHMiHMMHiHHMl-ll-ii-cNWflPINNNMWOONWNNPirO 



Q 



t^ OOO ro o> •* o> >o O »o I 



. rooo r-1 On •* OMO O vO W 



fO irivo 00 On i 



N •<*-vO fv On O N CO triNO 00 O 



(T) -^vO tv On I 






COf^COOfOfOCOfOfOCOfO< 



NO VO NO NO NO NO NO NO NO NO NO NO t^ !>• I 



rororocDOrororocoroi 



O N -"I-nOOO O 



•^nO 00 O N -!^NO 00 O N -^nO 00 O 



■*NO 00 O « -^NO 00 



cororororO'^-<*-'^-^-^ioioiom \r,\i 



O N ■*nO00 O <M -^nO 00 O N -^NO 00 O N -^nO 00 O N -^nO 00 O N -^NO 00 O 

M M H M M o N N M N rr>rororocO'<l-'*--<l-'^-<1-io>r>ir)io u^no 

OOnO iDMN O ONt^tr)'<l-(M H OnOO no in O N O On t>»NO ■«l-mH OOO tvu->-<J-ro 

H ro in t^ ON M M -"^NO 00 O N fOU-it^ONM o ionO 00 Q n -*no 00 On H ro lO t-^ 
nOnOnOnOnO t^r^t^t>. t^OO OOOOOOOOOOOnOnOnONOnOOOOOOmwhw 
000000000000000000000000000000000000000000 OnOnOnOnONOnOnOvONOn 

0\'<^OnO tJ t^oONinONO NOO roONiOHNO 0)00 -^ O inw t^roONiOM r>.ro 

■^ iDNO NO (^ r~.oo ooo^OOMMP^c^f^^■<J-T^lr) inNO t-^ rvoo <)0 on on O m m n 
ONONO^O^a^ONONa^O^OOOOOOOOOOOOOOOOOO►-<HMM 
N N W W N N N N N rOfOcOfOrocOfOcOMcOfOfOrocDcvirororocoocoro 



• ONin O IT) O NO 



. N r^Noo rooo 



lON'^ON'^O lOO iriHVO 



OOMMNOro-^^^iO uiNO NO t->. t^OO 00 OnOnO O h h n fOfO'J--*in idno 
00000000000000000000000000000000000000 OnOnOnOnOnOnOnOnOnOnONOn 
NC<NNN(N<NWNP1WN«C1C»NP1MNNC4<N(N)NPINMNNN(N 



. 


•<^NO rvONO PI -"MOt^ONO P« --MO t^OO O N CO l/^ t^OO O m O idno OO H ro 


5 


5; ^ ^ S! S ^^^^^ Sl^ %%%%. s; ^InB n8 n8 nS nS nS I nS 1 n&nS^I' 

rOforocoococororomfncomrorororofOfOcorocorororocorororococo 




0«^vO00OPJ;*'g00 0PJ ^NO^ O CJ^ ;5;nO .» N ^NO 00 « ;J;nO CO ^ 



FUNCTIONS OF A ONE-DEGREE CURVE. 



2S9 



O On t^vO lO •* ro w O O>00 t^vTiTj-rON m OOO t^vO \r, -^ ^ n o OnOO t^ xr, 
. M N ■^\6 OO 6 N •ivO t^O>M cou^t^o^M ro tj-vo oO 6 ( 
OOOOOOOOOOOOOOf — ■ 



r^ (T) OO N 00 ■* M t^ M OnO N OMrt h ( 



00 C3^O^0 " w N roro-^M" ir)vO vO t^oo 00 0> O 



I ro fo fO fo ro 



(^Onu^OvO « r^MO-^OvO H 



moo -^ O lAH t^NOO Tj-OMTIMVO 



00 Oi O O O 



. t^oo 00 O O O 



I fO CO CO CO CO CO < 



I CO CO CO CO CO CO I 



N C) CO CO ■* 1 

CO CO CO CO CO CO CO ( 



invO 00 O O N CO ■«J-<0 t^OO O I- N -^ lOvO 00 O O N CO -^vO t^OO O >-< N ■* tO 
r^ O co\0 O covD 0> 01 lAoo M irioo >- •*■ c^ o co t^ O covO O 



CO CO CO Tf -"^ -"l- U-) 1 



in\o vo vo vo t^ i 



00 00 00 o o> a> I 
OO 00 00 OO 00 OO < 
CO CO CO CO CO CO ( 



■<l-vO00 O <M -^vO ( 



O O t^M3 
CO -^vO OO 



OOO t^vO ■<*• CO f) O OOO vO lO ■ 

I r-. o 1 



CO HI O O t~-\0 lO •<*• N H 
N ■>*-'0 t^ O M CO lO t>. O I 

o 



o o o o o o 







t". CO OVO N 00 •* w t^ 

CO CO ■<}- -"l- m^O vO t^OO 00 

(cococococococococococococococococococococococococo 



Q 

o 


N 


c-> cooo ■* 


o 


lO vO 


M t^ PJOO 


CO 0-* 





lO H\0 


N t^ cooo ■* 


o 


inO\0 M 


t^ 


M 


M 


CO CO ■* 


-S- irivO vo 


t^ t^OOOO 


OOO 


„ 


M N N 


CO CO 


T^ Tl- iri 


m^ t~~ t^oo 


no 




















O N N 




Ci W IN 


C4 








COCO 


CO CO CO 


CO 


CO CO CO CO CO CO CO 




CO CO CO CO 


CO CO 


CO CO CO 


CO CO CO CO CO 


CO 


S 






























Q 







CO Tj-vO 


t^ O N 


COIOVOOO 


O M N 


^ 


in t^oo 


w 


CO -^vO 


t^ 


OO N CO 




O 
X 


„ 


^ 


t^ O COVO 


O COVO 


Of) lOOO 


K lOOO 


^ 


T(- c^ 


■* c>. 


O covo 


^ 


H vO O CJ 


in 








irjvO vO vO 


































u 


















a 


o 


N 


■<*-vO00 


O 


N -<*-vO00 O P) -^vOOO O 


N 


■^VOOO 


O N 


■<^\ooo 


n 


N -J-VOOO 


O 


S 



























290 



FUNCTIONS OF A ONE-DEGREE CURVE. 



O « -^^ 00 O 



N -^VO 00 
(SOWN 



0\ H ro m t^ CT» I 



O O09 t~»vO lO ■* 'i- CO N 



O OOO t^vO 



lOvO *0 vO vO ^ 



N CH W N M 



1 « CTi^ CO 0>\0 



CO m ro CO ro r<^ fo I 



ICOCOCOCOCOCOCOCOI 



lCOCO->J--.l-Tj--^Tt-T4-- 



0> -* in w vo 



100 -"J-O^ OOO -"i-OMD 



. r^OO 00 On o O 

_ . t^ r^ t^ t^oo 00 

icococococococororococococococococococococococococococo 



•^^ t^oo o O 



»A^ t>.00 M N CO ■<*- 1 



oooooooooo 



-* C^ O CO t^ O co^ 
T^ "^ IT) m lo^o vo vo 
oooooooo 



OOOOOOOOOOmhm 



O N ■*vO 00 O N -^VOOO O M ■<*->JD I 



•>l-vO 00 O PI -^VO 00 O M -^^ 00 O W 



O OnOO t^vO in CO N 



O OnOO t^vO 



O ON t->NO 

l-O 00 O 



.aNOvOOOOOMMHMHHNNC^NNCOCOCOCOCOTj--^ 
OOmmmmhwmhmmmhmihi-iwmmmmhmm 
0(NNOCJOP)(NOWPiC)OMNNC)N(N(NNNNC)<N 



t^ThO t^coO^O CO OinO CO 0\D 

t^OO 

NO NO 



N N CO -^ ■^^ IOnO no tv 
t^t^t^t^l^t^t^t-~t^ t^OO OOOOOOOOOOOOOOOOOOOOOO 

rococococococococococococococococococorococococorofocococococo 



NO H ts.N0O -O-OniOm t^ COOO ■* O NO 

inNO NO r^ t^oo ooonQOmmncoco 
•*T^■^l-T^■<l--.l-•<^■*mlnlnln^nlnln 

fOCOCOCOCOCOCOCOCOCOCOCOCOCOCO 



coONinMNO MOO •& O <n 

in lOND t^ t->00 00 On 



CO On 
(N* cj 



P) 

O I 

cocococococococo I 



00 On M P) CO Tj-NO tvOO O 
O CO t^ O COND On C>) in On 

U-INO NO NO NO t~~ t^ r ' 



inNO t^ On O 



■^nO 00 O N ■•t-NO 00 o 



FUNCTIONS OF A ONE-DEQREE CURVE. 



291 





! M 

1 ^ 


2 


N ^VOCO O CJ ^vooo O Cj ^^CO O CJ^^^M O ^^vO0O_O N ;j;^c» O 


< 


H O^oo t^ r^vo inin^roN <N h o Ooo oo t^ t-^vD i/^iOTj-roroN h m 


2257 

2259 
2261 

2262 
2264 
2266 
2268 
2270 
2272 
2274 

2276 

2278 
2280 
2282 
2284 

2286 
2288 
2289 

2291 

2293 
2295 
2297 
2299 

2301 

2303 
2305 
2307 
2309 
23II 

23^3 
2315 


u 
u 
en 


>0 rOO t^-«»-i-i00 lOD OvO ro 00 10 N CTivD -^moo von t^^HOOiO roO 


OOOOOiiOlNrOTl--)- invo l->. t^OO 0^0^0 m D M rOTi-u-, invO t^ t~-oo 
(N N r<^mr^o^roror<^romro^-)o^(-oror^Tt-TfT^-T^T^-Tl-Tl-^Tt-T^-^^Tj-u^ 


d 

Q 


t^roO"lw t^Tj-0>0 NOO -"hi-i t^rOCTiiAMOO -^O^O Ptoo u-ih t^momN 


11^8.0 ^nilf 5^1-1-^11-115-^^ ^^^^^^5^^^ 


Q 

i 


H pj ro Tf \n\o t^oo 00 H PJ ro I*- IT) lovO t^oo h w ro rt- u->\0 t^oo 


NNNN(NNININ(NNN(N(NlN(NNDD(NNNONtNC>JN(NW(N(N(N 

Tl■'<^■■.J-■^^T^-.i-M■T^T^■<J-Tj-^TJ--*•Tt-T^T^T^T^-TJ-T^T^Tl--*TJ-T^^^1i-1*••<*■ 1 

1 




« ^VOOO 2 I^VOOO PJ ^^<^ O^PJ^^VDC^ ^q:^0O.O^PJ^V^OO 


«* 

>* 


2 

§ 


<N ■'J-'O 00 N ^VO 00 P) ^^O 00 N ^^ 00 IS -i*-vO 00 P) -J-vO 00 O 

HiMMHwPiP)NP)P)roforo<r)M^Ti-^,i-^ininiOin invo 




iriTt-roroP) m O ooo t^vO u^-!t-T^mp^ w O O OiOO t^^o vo iD ■*• ro m P) h 


Om romt^ONw p« ^vo 00 M ^^ CO PI 1^^ t^o>M mu-)r^oH nmt^. 
OnO " w 1-1 H H P) P^ P) P) PI mmrororom^^TfThTHninioiD 
iHPiP)NP)p)P)P)P)Pipqp4P)NP4Pipqp)P)NP)P)Npqp)P)P)Pir<DP) 
NNP)P)P<C<P)P)P)NP<P)P)NMNWDPlPiP)P)PJP)P)r)p)P)P)P)pl 


1 
& 


t~»roO t^^HOO inp) Osso roo^ fOO t^-<J-MOO inN o\\0 MO t^ -* h 00 \0 


t^oo O^C^O H H PI r<irr)->i- invo VD t^oo OOOOOwPlNrOiJ-ir) invO t^ t^oo 


d 
O 

Q 

s 


VD NCO r^0vO PJOO ^OvO PJOO ^0\O PlOO tCOvO P)00 ^OvO PiOO '^O t^ 


Owi-iPirorOTf'*- xn^o vO t^ r^oo oso>0 w P< P) rorOTi-u-) lovO O t^oo 00 
oooooococococooooooooocooooooooo 0\ (j\ o\ o\ 0^ 0^ 0^ 0^ 0^ 0\ 0^ 0^ 0^ 0^ 0^ 


d 

a: 
o 
a 
O 


CJi 1-1 PI fO -"l- lA^ 00 H N ro Tj- lovo t^OO On M P» ■* "^^ t^OO CTi M 


^ 2 ^^ 2^S J?^ f^^^:^5:^S,??^S?^^vg^?l-lC'i^.So^c!5-aSS;8 




7; 


P) ■♦^80 N ■<t-^ 00 N -+^CO P) -^^Or^ P) ^^D 00 PI -<^VO 00. 


tKX.-=r^ 







292 



FUNCTIONS OF A ONE-DEGREE CURVE. 



O <S -^VO 00 O N -^^O oe O PI -^^ OD O D -^^ 00 O N -^^O 00 O N -*VO 00 o 

M H M M M f) M N ci (N cor'irororo-<t--<l--<l--<)--*ioioioio io\0 

•>j--<ff<lroP)WP)HHOOO O>oo 00 00 t^ t-^'O vOvO inmm-*-^-^r<irON N 

rriinp^O>>-i roint^Ow ro Th^ oo d N -"l-^O oo O N 'i-^ 00 O w -^^ o6 O N 
t^ t-^ t^ r~.oo ooooooooooONOOOOOOOt-iMwi-MpjNMDNroro 
mrororofomroc-orOfnrorororo-<t--*T)--*'>J--*-*-"^-<l---4--^'^-^->l---»--^^ 
WNNNMNtM<M(N«NOClN(NCMNCJN(NWCJPINCMMONNPl(M 

H 0\^0 ■* M On\0 ■* H On^O ■* M ovo ■*« Ot-^inN O P-.mro000 lOrOMOO 
p) P) m ■* m irivD t^OO OOO^OHWN^V)T^■^ lovo t-^OO OOOOwMPJrO-*-* 

t^t^r^i-^t-^t-^t-^f^r^t^ t^oo oooooooooooooooocooooooo oooono^oo 
■*-*-^-*-<l--*-*'*-^-*-4--<i--!t-*-*-^--^-<«-->*--<)--^-<J--<}--*^'iJ-->*--<j--<l---*--<*- 

MOO iDMOO -^O t-^roO-O (M OMTiMOO -^H t-^rOOMANOO lOM ^^■<^l-l00 "1 

vo vO r^oo OOOOOHNWroro -i-'iA invd t^ t-«00 OOCT.6oi-MPff<^-*-<fm 
r^^^ororor<^r<^-^T^T^■*■*■<l•■^Tl-Tl-T^■>J--»J--*■■*■^l-•*u^mlr)l^lU1lOu^u-)ln 

m>0 r-» tvOO OvOOMNrOro-* irivo vO t^oo a>OOHNroro-<t- irivo >0 t^oo 

inoo M -!t- t^ Q ■* t^ O rovo On n uioo w -* t-^ ■* r^ O ro^ C^ N looo m ■* t-^ 
0000 OOOO O O w w H M PI P4 w mro(-ri,j..^-^u-)vnui invo vo vo t^ t-^ f^ 
r(^romrofo-^j-■^^■^^-•*■-J-■*.^-.f•*-:^-^ ■*.■* T^->J-■^■^J-■*•-*•■^T)-■*■*■^^•<^- 
1*•■*■*■<l•-*--^-<^■■*■*-*-<^■■^■*■^'i*•■!^■*■^-<l■-<*•-r■<l-■^•«t-■^■<J-•<J-r^■<l-■*•^ 

O N -<1-^ 00 O P) -^-^ 00 O PI •^'O 00 O PI -"J-^ 00 O PI -^VO 00 O N -*->0 00 o 
1-1 M HI M M PI PI P) PI P) rororororO-^-^->i--*-*mioir)io ui\0 






PI P) P) PI rorocororO'-a--<l--<J--<*--<l-mmmui i/ivj 



H O O ONOO 00 t-^ t^^ ^Oin-<J--*rO(r)PlMHOOO OnoO CO t-^ t^^O vO tn ui ■* 

lA f^ O P4 ■'l-vO 00 O PI -<l-^0 00 O PI -*nO 00 O P) mint-^Ov'-. r^xot-^ON" "^ 
H M i-c M pj PI PI P) ^^f^^ro^^)!n■*■*T^-^Tt-Lnu->lrlLnu^ mvo vD ^ \0 vO t-> t-v 

MPip<pipiP)p)P)pip)PipiP)p;pipipiPipipipipjpipiP)pi(NPip<piP) 



s^" 


t-^iOPl Ot>.T)-M 0\^ rOHOO u-ino t-~u-)P) Ot^Tj-P) Ot^-*M ONVO -<t- w 




00i-PlPi(^"^iO in^D r^oo CO CT..0 m m pi ro ro -^ ulvO ^O l-^oo O O " N 


d 
O 




PIOO -^H t>.roq^ P) ONiriwoO -^0 t-^ro OvO PIOO triM t^-^OvD ro on-O <n 

t-^ t^OO OOO >-> " N PI ri^T)-T)- invO vO t^ t^OO OO^O m ■-'PI mrn-*T)- lO^O 
M«MM«rq(NPi(NMN(NP)P)Pip)WPIPiPiPirorOf^rororor^r»iroro 


d 


w PI P) ro ^ u^vO t-^ r--oo o M n (N ro '*- mvo t^ t-^co oOwPiPirOTj-io 



o o o o o o 

Tj- -^ Tf -*j- -"^ Tf V V -'i- W- W- W- V W- W* -^ -^ V V V '<t- ■^' '^ I 



FUNCTIONS OF A ONE-DEGREE CURVE. 



293 





°^ 




C ^.OOO O PJ ^^c^ Pj M-vOCg CJ^;5;0<« ^^NOOO p. JJNOOO O 


Q 


0>OnOiO>0>0 O O r- ■- -1 ►- M M N D M o forororO(-o->J-Thr}--«-Tj-u-) 
PINP)PlMP)P)P)NPICaNNP)NNP4NP)PlP)P)r4P)PlMP)P)P)(NP) 


X 

1^ 


rOOOOvO -*P) OoovO -<1-P1 GOOvO -"l-PJ OOO^o -^N OOOvO 'i-P) QoovO tK^ 


OOONOiOwPlrOrOT*- ino C-N t^OO ONOf->-p)ro-«-iO u-)\D r-00 Oi ON M M 
M M M C4 N Pt PI p< P4 PI p) PJ p) P) P) f0^or<^r^o^rof<^^or<^roromro^-<^Tt■ 


Q 

o 


N ONiOPJOO inpJOO lAM t^roo t^roo t^roO t^-*MOO lOP) O^ ro t^ -^ 


lO lovo P^ r-«oo OnOnO ■"! w PI rom-<j-i/-) invo t^ t^oo OnonQ " M D rOTj--*u-) 
rv r^ t^ t^ t^ (^ r-, t~,0O OOOOCOOOOOOOOOCOOOOOOOOOOOOO OnO'OnOnO>OnOnCT> 


a 

i 

u 


t^ t^oo oOONONOOwMpgpifTiro^^ir) invo vO tv t^oo OOOnOnQOmmPI 


ON PI ^oo M T^oo « 1^ 1^ mo ON P) inoo m ^ t^ o pomo on pj vn on P) iooo m 
NO r-N t^ f^oo 0000 OnonOnO O O w H i-i D M (N r^ro^om■<^-T^T^u-)u-l invO 
miOLOu-)u-)iJ^u-i LOin u~,\0 nOnOvOnOnOnOnOnOnOnOOnOvOnOnOnOnOnOnOvO 




p. ^NOCO O 2 JNOOO g ^vocg J^^vOOO O « ^NOOO.O p. ;5^vO00^ 

■ 


1 




« ^VOOO O 2 2"^^ PJ ^0<^ 0^« ^^WO^ 5^0 00.0^ C^^NO 00 o 


Q 


PJINwhwhOOOOnOnOn OnoO 0° 00 00 t^ t^- t~. t-.NO nOnOnOvO ininiOlAin 

pJ 4-vO oo 6 c^ ^NO oo ON- roint^ONM r^int^o-" roiAt^ONM roint^ONM 
r<^ror»^m■^l-Tl-T^-^1i-T^u^lOu^lr) idno vDvDnOnO t^t^t-^t^ r^oo oo oo oo oo On 

PlP)PtP)P)P)P)P)P)P)NP<PiPIP)WP)P)PlPiP)P)P)PINPIPIP)P<PIPI 


u 


OONO irtPi ONtxinroOoONO M-w ONt^inmooovO ■*- ONf^mrow ONf-iDro 


S; S ON S; S^'Sn S;8ooooooo'oo'o'oo2MM2M2'l?M'2Er'S 


555:^5:5:^S,S,S,5,gig,g,g,g,g>g,g>g>C^n^?,D^;;:,;;:,S^j:;,;;:,n^l;!, 


Q 

O 

Q 


lOMOo ThM ^>.T^o t-^foo t-^rooNO i^Ono co onno co onno p» onvo p) ONin n 


lONO NO t^oo 0OONOO^-lP^P^ror^^ vanO no t^ t~-oo OnOnQ •- " PJ rOrOTj-io 
in in LO LO LO in LO\D nOnOnOnOnOnDnOnOnOnOvOnOnOnOnO t^r^f^-t^t^t^f^ts. 


Q 
o 

a 


ooosONOOMWP)r<T*-* iriNO vo r- t^oo onOnOwhpjnm-*--* mvo vO r>. 


f-- m t^ rONO On Pi inoo " ^i- t^ o (^vo on pi no On n inoo M ':^ t^ ponO 0> 
r^oo 0000 ononOnonq o o 1-1 M M « N (N Pi fomm-^-^-^ioin iono no no no 


55 


PJ^NOOOO PJ 2-^<» CJ j^NO^ p^ PJ^ M;nO c« ^ ^NO 00 p. ;*vo 00 ^ 



294 



FUNCTIONS OF A ONE-DEGREE CURVE. 



z" 


n <S ^vD 00 M Tl-vO 00 


Si 


^^^ ^ To ^%% ^ ^ ^^^ ^ ^ ^^%S 


< 




in 


10 m .n ^, u^^O vO ^ vo 1-^ t^ t^ C^OO 00 00 00 a. 


26II 
2613 
26x5 

2617 
2619 
2621 

2623 

2625 
2627 
2629 

2631 
2633 


lOt^O-M romt^^OM r^^L^t-.o^" r<^u-)r^0^w 
romroT)-^-!(-Tt--rt-Lommu-) mvo vo ^ ^ t^ 

(NN(NnO(NO(NN(NNM(N(NN(N(NP)N 



. IT) ro (N O 00 



ONOO \o in ■* 



t^ r^oo ON O H n N ro ^ irivi 
irimmminioirjirjinu-jirii 



O m N OvO ro O 



u~i\o r^ t^oo o> O O M N (N (-0 

mhmmmmOMOJ.NNNi 

lOioiOLoinioiOLOLOLotOLoi 






cs 04 0) N IN 04 M rommroror^iroromro 
LOLOiOLommioiouiioioioiOLOinmio 






[-^ t^OO OOOO 0^0^0^0 O O O 



r^ i-^co 00 00 00 00 00 ( 



■*^ 00 O D -^^ 00 O CM ThvD I 



10 U-> in IT) lOVi 



M(NM04(NNNtM 



M rou^t^O>H romt^ONw roxot^OM 
iri iri m IT) m^ vo^vo^o t^t^c^r^ t^oo < 
iTi ir^ iTi vr^ iTi LTi \j~i w) \j~i iTi u^ iTi iry in w^ \D 



\o \o VO 



rOM o^^~^o■*o^ ooo^o mroH a> t^vo ■*« Ooo t^uiroN Ooo t^^iorON O 
p) ro ro ■* lOvo r~.oo ooONO'-CMNro-'i- m\0 t^ t^oo On O w N N ro -^l- in^ r~ 



-<)- t^ 


Tt- HOO 


Tl- 


H 


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m N 00 


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0) OnnO ro no 


fOO 


p~ 


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mvO \0 t^OO CO On 


8 


8 






^ 




^^§8^8^ 





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^ 


H 


ro ro 


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w 


M 
































(N W ro 


ro ro Tf 


-* 


in 




m'ONONO 


t^ 


t^ tvOOOO 


ON ON ON 








„ 


„ 


N C^ 


" 


ro 


ro 



w "^ f^ roND ON CJ inoo w -^ r^ O roNO on n 
NO NO NO t^ t^ t^ t^OO OOOOOnOnOnOOOOw 
nOnOnOnOnOnOnOnOnOOnOnOnO r^f^C^t^t^ 



•*nO 00 O 0) -^nO 1 



O 0) 'J-nO 00 O P) -"J-nO 00 O 



•^\0 00 O N -^nD ( 



rorororo-^-<l-M--*-^ini 



FUNCTWNS OF A ONE-DEGREE CURVE. 



295 



I lO lO LO lOV^ 



O H M N o ro I 

ro in r-> O I- ro i 
m ro ro ro ■* ■* ■ 
r^ t^ t^ t^ r^ t~» I 






r^OO 00 Ov o^ O 

a> I- ro m t^ o 



■*vO 00 O C4 ■* 



W IN N W C) 



O O t>»vD m "T i-o N O^ O OOO t^vO 



00 O^ 6 M IN 



OOCTvO'-'MN'^'l- invO t^OO On O O " D m I*- lAvO I^C 
M tH (N 01 N CI <N cs N CJ CJ w Oi rororororororOfOrommrn-^'^'^'^'^-^ 



Q 

o 

Q 


N Or^'J-MOO inroo r^-t-f) OvO rOMOO mpj o i-^^m On^O ro oo in cm on 


oooooO'-i-'Nm-s-Ti- in^ vo t^oo oonO w n n m-*->i- mvo t-- t^oo o O 
in in in\o \0'0^n£i\onOnononO'Ono no ^o t^t^c^t^t^t^t^t^t^c^t^c^t^t^ 
minininininminininLninininininininininininminininininininm 







^ mo o^ CN inoo w -^ t^ O m\0 o tN moo w -^ t^ o fo\D O 
I z; ^nro^o-*••^■^^lr)ln in\o \o ^ \o t^ t^ t^oo oo oo o on o^ o . 

' X O^ONC3NO^ONO^O^C>O^C^»O^O^O^O^ONO^G^C^ONONO^C7^0^0 



U-) u-> lo in IT) m 



O N '^^O 00 O <N '^O ( 



rorO'^-^-^-^-^u^iou^iA lO^O i 





N -^NO 00 O N -"fvO 00 N ^nO 00 O N ^vO 00 N ^nO 00 N -1-nD 00 O 

w „M M w (N N N N (N mmc^mcn•^T^^T^-*-lnlnlnln invo 


Q 
< 


ONON000i-iMwNNmmm-*-^-*in mvo vo vo t-~ t~~oo oo oo o^ o o 






■<j- N M onoo tx in ■* m w 0> t^vO in m w O Onoo t^ m ■* m pj w ooo t^vo m 
ininminininmin invo \dno\ono\osdndonovonovo'OnOvovOnO\0'Ovovo 


Q 

o 


00 in N o\vo m t-^ ^ M a^^ mo t^rhMoo mo o t^-*Moo mo ovo ^ n 


vO t^oo ooONOMMPjmrOTt- inMD vo t^oo ooONOwMomro-^in m^o r^oo 
mmmmro•*T^T^^Tl--*•T^T^T^T)-T^^T>-T^lnlnlnlnlnlnlnlnlnlnlnln 
ininininininininininininininininininininininininininininininin 


d 


N N N PI N rorommrommrfTj--i-r*-^^,f-*-»nininininin inso vo vo vO 


O 

5 


00*00 00 CO t»o? 00 XI OT co"oo"co"oo'oo 00 00 S'S'<S' a^o\o^a^o\o^o-cr^O'0'^^ 


z 

s 


f. rhvOOO 2 ;*VOOO O CJ ^vO« O fJ^^NOC^ ^^.OCO O p. ;j;<0<« O 



296 



FUNCTIONS OF A ONE-DEGREE CURVE. 



0\ as O w N rO-*ir) invo t^OO O O 
vO 00 H ro Lo t^ c> 
OO CO 00 



lOvO ^O VO ^ VO t-^ t>~ I 



t^ tvOO 
00 00 OO 
N (N D 



.00 CT> O 

o o" o'o'l 



^VO 00 O O -^VO (.. . , . 
100000 0>0>0>CT'00 O (_ 
1000000000000000 O^C^O^O^O^O^O^O^O^O^ 



t-^vO lO -^ -* ro N N 



O O On OCO t^ t^vo <0 



ro m ■<*• lovD 



(N ro T^ tnvO 
VO vD vO vO VO vO vO VO vO vO vo vO vo vO vO vO vO vO vO VO vO vO 



_ _ _ _ _ f4 ro ■* tnvD t^oo 

t^OO OOOOOOOOOOOOOOOOOOOO OvCT>OvO>ONOO>OSOva>l 
- ---- - - - -vovOvj- ---- ■ 



I vO VO VO VO VD 



Ooo loroooo loroooo loroooo inroooo mr^Ooo inrOHOO iritriooovo 
N <N m Tj- in invo f-.oo ooovOHHMro-*-^ mvo t^ (^oo ovoOHPjromTj- 
OOOOOOOOOOOhhmmhhmhhhmmmnNWNMNN 
VO VO vO vO VO VO VO VO vO VO VO VO vO vO VO VO VO vO VO vO vO vO vO vO vO vO vO vO vO vO vO 



O O O O OS O ON Ovoo 00 CO t^ I 



rovo Ov N inoo 



. O rOvD a> IN' lOOO H -^ I 



t^ t^ t^oo t 



uniommioiommi 



O N -^vo 00 o 



O P) -^VO 00 O N -!j-vO 00 O N ■ 



■>*-vD 00 O N -^vo ( 



t-»00 00Ov00HNNrO->J--ii- \DvO VO t^OO 00Ov0H(NNro-4-m yrwo t^oO On 

■*vo 00 O romt-^ONH romt>.o\H roiAt>.ONH ti-vo oo 6 <m -^vO oo O m -^vo I 

OvOvOvO O O O O M M H M M N N M M N rororoi ^^_^_^^. -.-.-. - 

r-. t^ r--oo 00000000000000000000000000000000000 



f) w O Ov OvOO t^vO m Th CO N N M O OvOO t^vO lOm-^fON f) H O Ov OvCO t> I 

VOVD 
VO VO vO ' 



•^"^"^•^-^"^iriirimiomi 



tr) M- lOVO t^OO 00 On I 



ro -^ invO t^OO On ov O 



N ON t>. ■<*■ H OvvO rOMOO inroo t^inw Ovr>-*N O 



ov O H ! 
«^00 00 O 

lo lo in 1 



ro Tj- in lovo r^ t--.co a\ o 



t^ t^ r^ tN> t-» I 



. t-. t^ t^vO vOvOvOvOvOvOvOvOvOvOvOvOvOVOvOvOvOvOvO in 



00 M -:}■ r^ O rovo Ov P) >ooo m ■* 
_ _ VO t-^ «^ t^OO OOOOCO OvOvOvO O 
uOOOOOOOOOOOOOOOOOOOOOOww 
inininininininioinininininininininininininioininin 



rr^^D 0\ CN inoo H -<*■ f^ o rovo 
PJ M p) rorom-^-^Tj-ini 



•<l-vO 00 O P) -^vo ( 



FUNCTIONS OF A ONE-DEGREE CURVE. 



297 



O N ->*-\O00 O N Tf^OO O 



Vi f^ -^ \r, t^oo O O 



O w M ■<*• "^^ 



Tt-vO OO O N -^^O ( 



o o o o o 



On 0> C3> OOO 00000000 t^t^t^t^l 

ON d 



t^ i^ (^ t^ 1 



ro ■>*- lOvO t^oo On O HI 



r^ t^VO NO VO NO NO VO NO NO 

On O H w r<1 Tj- lONO t^oo ON 

. .- .- .- in ID IT) 

t^ t^ t^ 



■^ M OnnO tJ- n O t^ in ro I 



CI O 00 NO ■<*- I 



tvOO 00 ON O 



NO t-^00 OOONOMNNrO-^in lOND r-^OO On On O 

to lO lO l/) tONO NO NO NO NO NO NO NO NO NO NO VD NO " 



NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO 



t^NO NO in m ■ 



in in lo tn m lo 1 



M -*• t-~ O 

N N N m 



ONOO 00 t-^ r~.NO NO inin-*-*mrO(N n m 

I inco iH T^ t^ o fONO On o moo m 'j- t^ o 
ifOcO'^-^'^minin inNO no no t^ t^ t^oo 

iinininininininininininininininin 



■^NO 00 O « -^NO ( 





« ^NOCO O PJ ;*NO<« O CJ ^NOCg CJ^M^NOOO^O « :5:nO00 « 5;nO<» O 


< 


inNO 00 On w N m Tj- lANO t^oo On w N <^ mNO t^oo ON o M N n- inNO i~-oo 


ON w n moo (N -^no oo P) -^no Onm mmt^ONM ro moo Pi -^nO oo o m 


i 

X* 


On onoo 00 00 c^ t^ t^NO Nommm^-^poforowNPiwMWHOGOOOON 


8ooooo'oo'oo'2l:;«!^MM'2£r'2 2'SSS^^J?'g^'SS^N' 


t^t^r-.t^t^r^t^t^i>.t>.c^t^c^ir^r-.t^t^t^r-.r^t^t^t^t^i^t^t^t^t->c^c>. 


Q 
a. 

o 

a 


NO -<*• M ONNO ->l-P) ONr^^Pi t^mp) Ooo mroOoONO pohoono -<i-h onno -<i- 


ij- mso nO t^oo onOnQ h p) mrort- mNO no t^co OnOnq m pi P) fn-5)-m mNO t-» 




t>. c^NO nono mmmT^T^^rommp^ pi pj m h m o q o onon onoo oo oo i>. t^ 


P) moo M ■* t^ mNO On p) inco m -i^ t^ o rONO On pi moo rONO On p) m.oo m 


2 


P. ^NOOO O p. ;^NO0O O PJ ^NOCg O ^;?;^<« ^^NOOO P. ;^NO00^^ 



298 



FUNCTIONS OF A ONE DEGREE CUR VE. 



D -^^O 00 O N •^^^0 OO O N -*>0 00 O C^ "^vO 00 O N -^O 00 O <N Tj-iO 00 O 

M M M M M o (N M w w rorororom-<J--*-^Tj--^u-)ioinu-) lAvD 

CO -<1-^ f^ o M N -^vO t^OO CM -^mt^oo N -^^ t-» Ov H ro •<j-vO 00 O w 

ro in r^ 0\ M -^vo oo O N -^t^ow rou-it^O P) -^^ 00 O ro lA t-^ O m -^^o 
M w M M N d P) N r<^roc<^^-J!-OTl--^^T^-!^l^lu-)u^u-) invo vO vO ^O \0 t^ t>. t-^ 

irofocororofororocorofocofocorncococorocorocororocofoforocofo 





P) N N rorocorOTj-Tf-^in irno \o t^ t^oo ooooONOOOMMNwroco-^Tt 


t^t^t^t^t>.t-«t^t^t^ t-OO 0000000000000000000000000000000000000000 


Q 
O 
Q 


'l-W OOO^O -^PI OOOVO ^N OOOVO -^Pi OOOVO -<1-P1 OOOVO 'i-ei OOOVO ■* 


sSslsss lllllllltllllllllUltir^ 


Q 


P) H O^OO t-^ t^vO inrj-rorON m O Ooo t>. t^vO >n-*^r<iPi m O O 


1 


00 w ■* t^ On P) moo H -"^ t^ rOvD On M m t^ COvO On P< lOOO w -it- t^ O CO m 
NO o. I^ f>. r-00 oooo o-OnOnO oo M H H N p) PI pq romm-d-^Tt-minin 

loiommmmmioiniomLommmmi/^miriminmioinmiomirjmioio 



P) ■<t-N£) 00 O P) , _ . 



P) Tf NO 00 O 



■ mNO 00 On I 



0^ O P) <^ UINO 00 On i 



ro m r^ On M m u^oo o 

IT) LO LO mNO NO NO NO t^ 

OOOOOOOOO 
ncororoforororom 



t^ t^ t^ t-^OO ( 
ro Ti- lOnO t^oo On O 



O O O O O O O 



romrorororomrorororO( 



OOOOOOOOOnOnOnOnOOOOhiwhN 

■<J- mND t-^OO On O M fO M- lOND t^OO On O 
I-^ (^ t^ t^ t^ t^OO OOOOOOCOOOOOOOOO ON 



ON t^ ^ PI O c 



N O CO NO ■* N O < 
ro ^ -^ u-)\0 t^oo < 



M O O ONOO 00 t^ t^NO in lo ■<*- ro ro N m 

O rONO OO M "^ t-^ Q rONO On PI moo m -^ 
OOOOCOCOOnOnOnOGOOi-'MhPIP) 

mmmmmmmmmmmmmmmu^ 



00 t^NO NO m in Tj- ro ro P) 

w ■'t- r~ rONO On n moo 
■tj- '^ -^ m in m m^o no no 

mmmmmmmmmm 



-^VO 00 O PI ■^no 00 o 



FUNCTIONS OF A ONE-DEGREE CURVE. 



299 



■*\0 00 O CJ -^^O 00 O O -"l-vO ( 



M -4-^ 00 O 



N f) W (N N 



ifirorOfOCirororoi 



I CO ro ro CD 



iO>0 t^oo CTi O O H N r<i 



I lO^O vOvO'O'OO^O^ vO 



\no r^oo CT. O " N ro ■* >nvo t^oo o^ O w w r^ ■* 

lAvO rvOO On " N ro -^ vnvo t^oo o o N fo ■* mvo 
ovoo^Ovo t^t^r-«t-^t^t^f^t^ r--oo 00 00 00 00 00 

0000000000000000000000000000000000000000 



00 'o ■<»- m M ov t^vo 

(S ro ■* \r,\0 VD t^oo 0> O 



O ooo t^>o in ■* fo N H o onoo 



rovo 00 M •*• t^ rovD _- .-_ __ 

-t -^ ^ irun irivo MD vO ^O t^ t^ r-~oo oooooo oononO O O 

inu-jin" * " * "" ■" "■ ■" " "" '- " " '- '- 



fo^o 0\ N tnoo 



inioioiomiotommmmmmi 



moo M -* t^ o 

M M D N PI rO 

minmiomvommioiom 



•0 f) -^^oo O 



O N -^^ 00 O 



w CO irj pv On I 



CO in t^ Oi 



t^ t^oo 00 o 
mromc<immoocnror<icocni 



rOfor^mrororooorocoro 



■* in invD r^ tvoo Oy 0\ O n h n rot^-^Th mvo vo t^oo OnQ O h n fo-*ini 
M N m '^ invO t-^oo On w 0) ro ■* invo t^oo on O w N m ■•J-vo f^oo On O m N i 



ro m ro CO ro 



m^<^-^T^Tj-■^T)-T^-!^TJ-• 



ioooooooocooooooooooooooooooooooooooooo< 



-^ N O 00 NO 
00 On O O H 



OoonO inrOM o>t^ 
inNO t^oo CO ON O 



ON t^vO Tt-IN OOOnD ■*« O OnOO 



C)P)(N|PJP)CJ(Ni(N)0)NM 



M ro ■* invo f^ t^oo On O 1-1 H M 



Onco t~.NO in ■* r<^ N N 1 
inoo M ^ t>, O ro\0 ON ( 



Onoo t^NO in ■* m N H o onoo co f-vo in ■* ro n h 

t^ O rONO ON N moo 11 -^no On n inoo 1-1 Tj- t-~ o ro 
"*(>qNONONOOO>:<i:^iHiHPiNCjrororf) ■ 



« ■♦no 00 o 



300 



FUNCTIONS OF A ONE-DEGREE CURVE. 



O <N 't-VO 00 O CJ Tl->0 ( 



Tj-MD oo O ro I 



I r^ O N -^vo ( 



t-~ t^co ca ca ca o^o^oo^c^o 



O O M H H 



i LOCO O N Tl- 



rocOcOfOrOfOfOfOcOi 



■ m m r^ ^ -^ 
Tj- Tj- Tj- ■* ■^^ 
I ro CO ro ro m 



io\o oo CTi H <N ro invD 00 Ov I 
0) ro Tj- LD t^oo 



N Tj- in f^oo O P) ro invD oo 
ro -^ mvo l^ O O 



. O M N ro m>o t~~00 O O M ro ■* 
(NCMN{N{N(N{N(N(Nrororororororororo-^Tj-TfTj--^T^'^'^Lr)ir)io 



On t^vO -^rOM 000 t^mT)-(N H OvCO VD lO rO N O O t^\0 'i- rO H O 00 t~«. in ■ 

ro ro '^ in^o t^co o^oo w w ro*^in ino t^c 
. _ _ _0OOOOOOOOmh«mwmmmh, 
t^oo ooooooooooooooooooooooooooooooooooooooooc 



M ro ■* lOvO t^OO CX) o ( 
00>OnO>On<>000( 



in t^ O ro^ 



H osoo t^ in T)- ( 
in t^ O rovo On ( 



ON03 t^ in -^ 



) NO 1^ i^ r^ r^oo 00 00 T^ -iv qn qs q 
icooooocooooocooooooooooo on 
"iinininLntnininininininLnin 



<M -^no 00 O 



O N -^no 00 O « -*no 00 O n -^no 00 O tM tJ-nd 00 O n -^nO 00 O n tJ-nO oo O 
MHHHH^lC^N0^(Nrororororo■*■5^■^rt-■^lnLnlOln inNO 

N ■* t^ ON H rONO 00 O ro in t^ On P) tj-no on h roNO 00 O ro inco o N in t^ o n 

00 O CNl T^ t^ On M rONO OO O M ■* 1^ On M rONO 00 O w in r^ On M ^NO 00 O ro in 
o rt M M M M N N N N rororororoth-*-*rj-inininin u-ino no no nD t^ t-^ t^ 
rororororororororororororororororororororororororororororororo 
rororororororororororororororororororororororororororororororo 



i 


^ -j^ tN.00 ON H ro ^ inNO oo' ON w tM ■* inNO t^oo o " ro t)- in t-^oo o i- n 




OOOOOOOOOOCOCOCOCOOOCOOOOO OnOnOnOnONOnOnOnONOnOnOnOnOnOnOnOnOn 


d 

O 

. 

Q 


t^ in -^ N HI ON t^NO -4- ro H ONOO NO inrOM Ooo i-^inroN O ONt^inTj-w m a- 


nOnono f^r^t^t^t^t^t^t^r^t^i^ t^oo cococooooooooococooooo onOnonon 



O ONOO NO in ■* ro N O ONOO t->ND -"l- ro w h O 00 t^NO in tJ- N h O onoo no in ■+ 

^ O rONO ON i-< 1^ t-~ O rONO On N -^ t^ O rONO 
i r^ t^ t^ t^oo ooooOnononOnOOO >-►-.►. 
-r^t^t^r-^t^r^t^i^t^t^ r^oo oo oo oo oo oo 
■^LnininininininininininininLnininin 



O PI -^No oo O 



FUNCTIONS OF A ONE-DEGREE CURVE. 



m 





O CJ TJ-^OOO O M TfvOCO O <N -^^co O N -"h^OO 
HMMMMMWNNNrOCOrOMCO 


^^5^^0S.;5;vCCOO 




tr,\o a\ (S moo m ■* t^ o rovo on n moo h -.j- h, o 


rovo O rovo On N moO h -^ 


M ro moo O N m t^ ON N ■<^vo co w co moo com 
MHMw<N(N^NP^(NroforocO'i■T^-n-T^mmm 
mmmmmmmmmmmmmmmmmmmm 
cOfororococncororococorororocorororOMro 


t^ On c^ -^nO 00 m r<i moo O 
m m^o NO NO NO t^ t^ t^ t^oo 
m mm mm m m m m m m 
rororororomrorOfOforo 





romt^ONH romt^ONM 


CO m f^ ON H -^NO 00 o w m t-^ ON H (dno co o (n •* tx 


X 


O w N m lONO t^oo On m 
OnOnOnOnOnOnO\OnOnO 


(N ro T^ m t^oo ON o N ro Tj- mvo 00 On h ro T^ mvo 

8888888oS55S55oSSSSSS 





OncO t^NO -* ro I 



mvo t^ t^co ON O M 



(T) -5)- -* lOVO t>.00 ON O M M M ro ^ mvO t^OO 00 On O 

-TT'T'-^'T'V'^'^w^Lnu^ioiommiomm mNO nonononovonononovo^no c^ 
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 



oovo iriroH 000 rvioroN Goo t^mrof) O ONt^m->^N O On t>.NO tj- n 



t^ O rONO ON N -^ 

00 ON ON ON ON o o 
OnONOnOnOnQO _ _ _ _ 

lo lo lo m mvo vo no no no no 



T^ t^ O n-) moo 



mNO On 1-1 -^ f> O rOND 00 

M M H N fJ CNi rorororo-«--<r-^mi/im 

ooooooooooooooo o 

~ ~ VO \Q NO NO NO NO NO NO NO NO NO NO NO NO 



NO NO NO NO NO NO 



O N -^NOOO O M '^NOOO O 



On M VO p^ O fONO 00 H ■<i-txONN mt^O fONO ON N m tx o rONO ON e) iTioo O ro 

N >0 O. On N -^NO 00 H rOlOt^O N -^t^ONw rONO 00 O (O m t>« Ov W -^nO On m 

■^ -"j- -* ■* m >n m uivo no no no r^ t^ t-. c^ t^oo oooooo OnonononOnO O O O h 
-^■^Tl--^•.J-T^■^1^-l^^Tj-Tj--.j-•>J--*T^■.i-■^•*■.J-■<J-T^•i*•1*•■^-*mmmmm 
focotofocococororococofocorococncorocorococorocororocororoforo 



O N romt^ONO N •<i-NO 00 on i 



fO mNO 00 o (M -^no 00 o N -^mt^ONH co 
D ro -4- mNO t^ ON O 



-rr^O t^OO ON O M ro ■* iriNO t^OO O w N <0 -*nO t^OO . . 

-mmimmNqNONONONONONONO t>.f>.rxt-.t^t^t-.t^ t^oo oooooooooooooo on 
" iOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOn 



OnOnOnOnOnOnOnOnOnOnOn( 



•* f) H O 00 t>.NO •<*■ fO M O On txNO •* fO N O ON tvNO lO ■>*• fO h O 0» t^NO lO ■^ 

OOONOMHWfO-^ <ONO tx t^oo ON O H N CO ro •>*■ U1NO t^OO Ov O O w N ro -"l- 
w M N w N f) N N M N CJ N N <N ^0(OrO^OrO^OCOn^^Om^O■>*■■*M-■*'!l-T^ 
QOOOOOOOOOOOOOOOOOOOOOOOCOOOCOOOOOOOOOOOOOeOOOOOOOOOOOOOOOOOOO 



ONOO NO m ro N o ON tvNO •^ fO M ONOO NO m ro N O on p^vo ■* ro i 



looo O rONO ON ( 



. O rONO ON (N 



O rONO On w m !-» 



■ m m mi miNO NO NO t^ p~ r^ t^oo oo oo 



■*NO 00 O N -"f-NO 00 



302 



FUNCTIONS OF A ONE-DEGREE CURVE. 



z 

s 


« ^vooo o cj 2-^00 o gj ^vow o « ^vo« ^^^voco o « jjvoeo^ 


Q 

< 


'i-OO H lOOO N inoNvO ON-O O rovo O -<l- t^ M -^OO N ir> O fO'O O fO tv M 


O « »0 t^ 0^ N -^I-VD O M ro^ 00 r<l >nOO O N to t^ 0> M ^VO CT. M 'i-vO 00 t-l 



O 0) lOt^O N "3t>.0 NVOOO 



Th t^ O N >J^oo 



VO \0 vO VO M 
O O O O I 



I O H P) ■* uTO t^ 0> O H ro Tj- invO < 
f-> t-> (^ t^ t^ tv. t^ t^OO 00 00 00 00 OO ( 

OOOOOOOOOOOOOO 



■<J-\0 0% N tooo O rovO Ol N 

o 



g'g'S'o O O I 



Q 

o 

d 


ro N H ooo t>.vo m Tf m N h o CTioo r^vo m -^ M (M n o o>oo t^vo tn -^ m 


000000 o^O^C^O^O^O^C^C^O»C^O»O^C^C^O^O^C^C^O^O^O^O^O^CTlO^O^O^O^ 
• 



lOrOM OM^iOfOH OtxiOrOH 0>t>.tnrOH OOOVO •<*-Cl OOOvO "<*-N OOOvO 



O N -"l-vO 00 O M 't-vO < 



■*t^O M-i-nO C0t>.0 mt^O 



O N »0 t^ Ov N -^-O CTi 
OOOOOOOOOO OCTiOlO 



ui t^ O w •<*- t~. Ol 1 



rorocororofororococococoi 



t^ O N ■<*■ t^ OM 



■»^^0 On w rOvO 00 m ro moD O roiOt^O N lOt^O M lOt^O 

ON O H C» "^ invO C^ ON O 



t^ Ol O •"• P) ■* lAvO 
OOOOOOOOOOOOO 



rj- rnio t^ 0> O H N 
OOOOOOOOOOOOO 



f^ 0» w OOO t^vO 



o> t>.vo lo T^ I 



O ONOO t^vO IT) •<*■ <*> 



■<»- \n lovo t^oo ON O w N ro fO -"h lovo r^oo ct> O h w M ro 



, r>. t^ p~ t^ r>. t^ p~ I 



ONOnO^OnOnOnOiOnO* 
T 00 00 00 00 CO 



Q 


0>t^uT.fM OOOVO >orow O. J^\0 ■<♦-« 000 t~,iArOM OnOO v3 -* o o o> r^ «a 


O 

u 


N u^OO M •* t^ On N u^OO M rOvO On N lOCO O rovo ON N •»!- t^ O rOvO On m ■<*- t^ 

VOVOVONONONONONOVOVONOVONONONOVOVOVOVOVOVOVOVOVONOVOVOVCIVOVOVO 



FUNCTIONS OF A ONE-DEGREE CURVE. 



303 





"»^ 

« 


•i 


O N VvO 00 O N ^^ CO N -^^ 00 O N -rvD 00 O N "-l-vO 00 O N M-vO 00 O 
H « M « M M N N N (N (DCOCOrOrri-^Tf-^^-^lOlOmiO invfi 

1 


< 


vO ^CO N vD 'i-OO IN M3 T^00 NvO ^Oror^M inov Thoo N ^O m O 


IN ior-~ow -^i-^cTiM T)-\o O M rovO 00 « ro irioo Or<^lOt-~OMlAt^ONr^, 
OOO^OO O M w w w N N N N r«^mn^^-)T^-^■>^T^u-ll0^o invo vD \D 1 

oorororomrororoi^fOroi^rorororororororororororofnrororornroro 

1 


u 
[I] 
in 

W 


lAOO M T^f^« "^t^O rot^O -1-t^M T)-oo M inoo n in o M \D O rovo O rovo i 


« (N 4-ui^oo ON c3 rn 4-^ t^oo (D - (N Tj-mooo (> 6 IN ro 4-\0 t^ OMD -i 1 


Q 

O 

Q 

i 


oo t^-O inio^mp) N M O ooo t--vO vOm^roroNnOO O^oo oo t^vo vO i 


M P) fO -J- irivO t^OO On " '-' N 1^ ^ mvO t^OO On >- (N CO -J- ■*■ lOvO t^OO ON 
O^O^O^O^O^O^a^O^O^O^O^O^a^O^O<O^O^O^O^O^ONO^O^O^O^O^O^O^O^O^O^ 


§ 

K 
(J 


N O r^inroooovc ^w CJM^^w t^inrOHOOvO tJ-m OnI^-^w 000 \r, m 


U1O0 rONO On w ^ r^ N u-ioo M Tf^ On fj u-> t^ fONO oo •- ■'^ r-. N moo 
P) IN r<^roro^<^Tl-Tl-T^u-|vnu1 invO no vo vO r^ t^ r^oo oooooo OnOnOnO O 

vOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnO nO nOnOnOnOnOnOnOnOnOnOnOnOnOnO ' 




j 
O N Nj-NO 00 O N -<1-ND 00 O (N) ^NO 00 O IN -J-no 00 P> ^nO OO IN -<-vO 00 O 1 

M M M M M N P) w (N o mmcnmm^^^^^ir>\ninxn mvo j 


i 


2 


j 

N ^nO 00 N -J-nO 00 O <N ThNO OO N -*NO 00 O W ^vO 00 O N ^VO 00 

i 


Q 
< 


M T^oo N NO On m f^ M Tt-oo nnd ONrot^>-i moNi^t^o -^oo w no -^oo n no 1 


1- m moo N mt^o n -"i-r^ONM ti-no On i- roNO co " ro moo 6 i^ m t-^ 6 n i 
D « w (N rort^roro^^Tr-<^-^mmm mNO no nD no t^ t^ r^ t-^oo CO 00 oo On On j 


i 

in 


(N moo w 'I- t- rONO On n moo m T^ t^ o mnO On rONO ON N m On N moo M m 1 


2 2 2I 2 1 S S ^ ?n^ Z^^ H E'H'S'l's f ^ ^ ? ?^ S^'?^ 2- H- ^ 


Q 

Q 


ro N « Onoo 00 t^NO m -^ rn ro p) « onoo 00 r-NO m Th ro m m h onoo 00 " 1 


■>a- mNO f^ t^oo On ►- <N ro T^ mNO r~-oo 00 on m w m ^ iono t~.oo on on O 1- 1 
IN P) N p) PI N PI mror^popomror^rr)P^roT^-J■Tl-Tl■TJ-T^-^^T^^•^mm | 
OnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOn 

! 


g 

o 

5 


1 

■ 1 

NO ■<^Pl t^mrOM ONt^mroQoONO 'J-P) Oco rnt^M ONt^mrOMOONO ^J-P) il 


w -"l- t>. P) moo M rONO On PI m t^ rONO On m -4- t^ Q pj moO m ^nO On P) m> : 

rf -^ 1^ m m m mvo no no no r^ t^ t--.oo oooocoonononOOOOwmwi-pjp) 
PiP4P)P)PtP*PiPiP<NPiP)PiP<P)P)P)PJNNP)rorommmforomroro 
NOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnDnOnOnOnOnOnOnOnO ' 

1 




ON ^NOOO P^ 2-^~ p. ^^~0 ^^^~-0 ^^NOOO P. ;J;nOOOO j 



.304 



-U^^CTIONS OF A ONE-DEGREE CURVE. 



'1 


•£ 

S 


O M ^NOOO 0. 2-^00 CJ ^vOCg O « ^vow O N ^.DCO « ;J;nO00^ 



< 


H NO <n ON Tj- ON cooo o<t^eiNOHioO>nO-<i-ON^ON cooo rooo N t^ 01 tv. M 


00 romt-^o oi loi-^o oi int^o oi mf^o oi rft^ONOi ^r>.ON0i -^t-^ONOi 


i 


ONroc-,iH iooNrot-.H ^<y^<^<^^^ '?9'?*ri "^TT^I "C^T^I V° 

0) -^ lO t>^00 On w 01 ^ IDNO 00 ON w 01 ro LOvO 00 ON 01 ro mvO t^ On 01 rO IT) 




Q 

OS 

o 

Q 


t>. b>.NO NOu-)inTj-Tt-rr>rON0iMH0OON ONCO oo t^ i-^no vo mtn-<l-^fororo 


e^oo ON M 01 ro ^ m^o l^oo on O h oi oi ro -^ lovo t-^oo on O h oi ro -^ vond 

ooooooooooooooooooooooooo*o^o*?o'?? 


Q 

% 


MOOvO rOHOO iriroooo inoi t^inw ONtv^oi OnvO Tf h OnvO ro w oo no ro 


H rOND On 01 ■* t^ ro looo w t)-nO On 01 Th t^ Q ro inoo m -d-vo On 01 vO t^ O ro 

o^ 5: g^ 5 o, - ;?, - ;;, S!, S!, si> s> 2? s s ?; 5; ;?; s; K ;;! IS s^^^ f;^ s^ 

vo NO ND NO NO ^ NO NO "O NO M3 NO >0 NO NO NO 'ONONONONO NO NO NO NO NOVO NOVO NOVO 


» 

§ 


« ■'l-NO 00 N ^NO 00 N ^NO 00 01 TfNO 00 N ^nO 00 W ■'J-nO 00 O 

„ „ „ „ H ca M 01 M 01 fOcorororOT^-Tfr^T^^u^lo^Du^ iono 


i 




N -"l-NO 00 N -"l-NO 00 01 ^NO 00 01 ^NO 00 N ^NO 00 O N ->l-VO 00 O 
H « H M w d N 01 01 01 crirOrOrOOOTj-^^rf-^lOUIlOlO lONO 


Q 
«5i 


ONrOt«-MvO >00» rooo N vO h 10 On t»-00 rOt^«vO h »nO •*'0- rOOO « t«. H 




NO '*■ t^ H inoo 01 vo ON ro t>. -*oo H inoNNvo O rot>«H uiovf^t^n inov 


.^c^c^-c^cS'^cg'oNSNSSf^g^S S ?S-| S-8^2 S ^2-^^^ 8 ^ | 


d 
O 

Q 


• 

voio-*-*roo)oiHOO o>oo 00 r-vo vOlnTJ--<^roo^o^MOO onoo oo r-* t^ (■* 


ScgcScS<S'o?^S<^g§g§<g^8N3NSNSS;S^5;'§^g8 S S S S'og'l B~ 

OnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnO O O O O 




d 


ro H 00 NO ro H onnO ■<i-M ONr^Tj-ci ONt^iow t>.tnrooco mrOHOONO rOM 


VONONONONONONONONONOVONONONONONONONONONONONONONOVONOVONONONONO 




O M '!^NO00 0) ^VOOO O 0) ■'J-nOOO O 01 -*-NO00 O M ""l-NOOO N Tl-NOCO o 

HHHMH0icici(NC<rororororOTi-<^Ti-Tf^io«oio.o tovS 



FUNCTIONS OF A ONE DEGREE CURVE. 



30a 



2 

S 


N Ttvo CO O N -^^ qp 


g 


01 T^vo 00 o 01 
d N 01 0) m ro 


^^<«O0^^vOC«O0,;J;^0OO 


Q 


H t-> 0) r- 0) oo fooo <^oo 


Th 


o\-<J- o ■* O in 


OvO M t^N r-. moo M- o>-* oui w 


^'^ R'R'R'R'S 22"2 


2 


•<l- t^ O 01 in t^ 


Ooiint^OMint-^OOi inoo ro 
mmmm-^-^-^rt-minin mvc vO 


< 


o ^ ^ ^ ^ ^ :; ^5- ^ ^ 


■* 






u 

CD 
X 


Tj- o> rooo N fs. 01 vo H in 





'^^ 0^ ■<^ O ^00 


rooo rooo oi t^oi t^:: t--oi i-^oi (^ 


'OOM^H^^^^'P.H 


^ 


^^H^^^^^^HH HI; 1 ^ ^ 











mmOOOOnOO 003 00 00 CO t^ t^ t^ tvvO vOvO »ninin>n->i--*-.i-rorrirorri 

invO t>.00 oo CT> O M 01 ro ■* invo t^oo cy> O " 01 ro ^ in\0 t^oo o O ►- 01 ro ■* 
vOvO^o^^o t^t^r^t^i-^t>.t-~t^r^ t^oo oooooocooooooooooo octvOOO 
ooooooooooooooooooooooooooooooo 

0\\0 mO t^-*MC0vO roo t^-<t-MOO ino) OvD ro O t^ ■<^ oi 0\0 ro O f>- -!f >-i 

Tt- t^ O ro inoo w rovo o oi -"i- r-^ o oi moo O rovo On w ^J- t^ o 01 inoo O roo 
in in'O vo vO \0 t~. t^ r^ t^oo ooooovONONONOOOOi-''-'i-'-'Oioioir»-iror-> 

VO VO *0 ^ ^ VO VO vO vO SO "O vO OvO^'O'O'OO'OvO^^^OvOOvO'OOvO^O 



O N -^VO 00 O M ^vO 00 O 0) ThvOOO O 01 -<J-vO 00 O M -^vO 00 O 01 -+^0 00 O 

M HI M M w d 01 01 01 01 corommro■^T^■^■^■^lnlnlnln m^o 

N -"l-^ 00 O 01 -^-vO 00 O 01 -^vO 00 O 01 -^vD CO O M TfO 00 O 01 -^^D 00 O 

M M M H H 04 01 01 01 01 rororomrO'^-^'^'^'^inininin invfi) 

H vO M vO w vO M vO M \0 HVO H\0 ^ ^G M\0 MVO MvO HvO HvO w vO w vO •-* 

01 ^t^OOl -^t^O^Ol -^t^OOl --J-t-^OOl T^t^O^01 T}-t^O01 ^t^CTin ^l^. 

M « M M 01 01 01 01 mmrom-<l--^-^-<t-inxnm invo vo vo vo t-^ t^ r^ t^oo oo oo 
ooooooooooooooooooooooooooooooo 





■* On rooo 01 vO H in -*oo ro t^ 01 vO tn Oi ^oo rot^oivO w\0 O ines-* 


^1 if ^ ^ ? ^I-I- ^1 1 'f'f 1 1 f 1 i^ g^ g^f 1 i^ ^ H ?!H> 


M 




OS 


coroc«oiMMHOOONO> o\oo 00 b> t^ r^\o vo miom^^rororooi oi « « 


o 

a 


\0 t^oo Oi H 01 ro ■* ■<*• invo t-^oo Oi ►-i oi ro ■* ino t^oo on m n ro ^ lo 
roromrOTj--*-.l--^-^-.*-Tj-^Tj-^^minininininininin invO vo vD vo vO vo 

ooooooooooooooooooooooooooooooo 



p 


ro i^ in 01 ON^D roMOOinoi Oni^'^moo inroo t^-*M onvO ro O t-, m 01 O 


o 
X 


rovo 00 M ^<o CT> 01 in r^ O ro moo h ^md On oi in r^ O ro moo m -!j-\o on oi t)- 
mmminininmmm invo vovdnovonononovondnonononovonOvdvOnonono 

VOVONOVONOVOVOVOVOVOVOVOVOVOVOVONOVOVOOVOVOVOVOVOVONOVOVOVOVO 


2 


01 -^nO 00 01 -"J-nO oo 01 -1-vD 00 01 •^\0 CO O 01 rj-vo 00 01 tJ-xO 03 O 
H M H M M 01 01 01 c^ 01 rororororOTj-^^n-^mmmm mvo 



806 



FUNCTIONS OF A ONE-DEGREE CURVE. 







O N •'1-VOOO O N •<*-^O00 O N -"J-^OO N "^VOOO N -^^ 00 O W -^VO 00 




< 


OvO M l>.rOOMnH t>NOO -^OVO MOO -i- OO MOO T)-0^O NOO -<^0vO MOO 




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H 'O PI t>. M 00 rooo -^CTiiDHyD P) t^rocri-*0 mn t-^MCO roomOvO m t». 




00 O H pj ^ lo t~,og H ro inMD oo cy. h pi ■<^^ t^ on O n ro in^ oo m ro ^ 








p 

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osoo oooooocooooooooooooococococooooooooooooooooocooooooooooooo 




^H"s1 s''^ 2 &E'S^S^H'^^^'^^?3-H-HS^^2^1-r^E'E>S^ 




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vO ro O^O rOOvo rOO t^roO t^roO t^-<J-0 r~.-^H t^Tt-woo -^HOO tJ-mOO 






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« ^.OOO O M 2-^00 O PJ ^^Cg.O M ^^C« O M ^^OO O M Jj;^00 3 




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M vo p) t^ rooo ■* >n H vo c» t~s cnoo -ij- o tn h t-^ rooo -.t- o lo m t^ rooo ■* o 




ro moo ro looo m rovO <x> h m-JD oo h -^vo 0\ h -^vo CT<P) tj-i^op) -^t^O 
vD -O vO t^ t-^ i>» t^oo ooooooa^O^O^O^OOOOwHMMP^p^P^Plcocoro•* 

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t^p) r^P) t^P4 t^P) t^P) r^M t-^M t^N t^Pi t-pjoo rooo ■^on-'I-O movo h 




P) Tf lO l-^OO O H ro -^vO t^ On M ro inyO oo 0> M P) -"J- lO t^OO O PI fO M^^ 00 

m Lo u^ in lONO vovdmdndvovo t^t>.i>.t^ist^ t~-oo oooooooooo o a^ a> as o^ o\ 








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rOM P) M P) P) H H M H H H O OnOnOnOnOnOnOnO\0>0\OnOvO\OnO> 




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H 00 lO H 00 in PI O^NO P) ON^O roo t^roO I^-^hoO -<)-h00 mP) omom on\o 




NO 00 M -^-KO ON Pi >^ t^ P) moo M roNO Onm Tft>.ONM int^O ro moo m rONO 
roro-^-*-*--*mu-i m\0 vo vo nD t^ t^ (^ r^oo ooooooo^ONO^OOOOMHM 
t^t^t>.t^r^t^c^t^t^t-st^r>.r^t^t^i^t-^t^t-^t^r^(^t^ t^oo oo oo oo oo oo oo 






O W •'t-NO 00 O N -*NO 00 P) '^-vD 00 Pi f NO 00 P) -J-vO CO M ■<^^0 00 
M M M n M p) p) M P) CI rororororo-*--^-^-^-*-»nuiir)io mvO 





FUNCTIONS OF A ONE-DEGREE CURVE. 



m 



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N ■<^^oo O P) -^vOOO O N -^vOOO O « -^vOOO P< -^vOOO O P) •♦vOOO 
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t^Tl-H t^TfO t>.roO i^roOvO (Tiovo MOvo roo t^r^O rvrt-O t^-<fi-i tx 


mr»>'>J-^-*r^T^-^■*■<^-T*•■.J-■^^-<»•r^^■*•■*rl--^,J-,^-T^T^■!^-■^T^1J-■^-.l- 


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iDM t«»roOvO NOO -*0 t^fO O vO roOM-'5P>00 mn t^-*o t^fO OvO P) OM/I 


P) Ti-int^oo PI rom r^oo w ro invo 00 h m invo 00 o " M -fvo oo c^ h 
00000>OOOOOOi-iMMMMi-ipipipipipipqromrororor<-)rOTj- 




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M w M H H N N N N N rororornro-*-*-*^rrinininin invO MD vO vO \0 t>. 

4 in^o 1^00 o\ d H N ro 4- invd t^oo 0\ d H IN ro 4- in^ t^oo d> d w (N ro 4- 
OOOOOOOOOOOO a^O^O^O^O^O^O^O^O^O^O O w " m h m 


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T*- O t^ ro Os^ N OMTiHOO -^0 l^ro 0\0 N C3MOH00 '*-0 t-roOvO PI OMn 


o> oi'oN o'on'S. So^o^8oooooooooooo'?o'o'o'ooo'oo' 




N -+VO 00 PI -^VD 00 PI -i-^ 00 PJ '^^ 00 PI -"t-vO 00 P) ^^D 00 
„ „ M M M w PI PI p< PI rororororo^^-*'a--tl-iDiovr)io lovo 


1 


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O P< -"l-vOOO PI ^vooo P)-T)-\O00 O PI -^vooo O P> "^^00 PI ThvOOO 
M M M M M PI PI PI PI PI cororororOTi-Tf-i-Tt-Tfioininin u^\o 




00 in M p^ ro OnvO Nop -"l-O r^roONi^Hoo ■*>-■ t^roO'O roompioo iam t^ 


llllllilllilllililllliiMllllil 


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t^PlOO ■<^OvO H t^roOMOOvO P)00 ^shoonn I^Tj-OvO PJOO -*0 t>.roOMC 


■*vo t>. o M PI ■* »o t-.oo Pi ro m\o 00 OS M ro -^^ oo a> m m ->j-'0 t^ o o pi 
■*--*^-^mmminm mvo "OvOvovovovo t-^t-.i-^r-~t^ t-.oo oo oo oo oo oo os o> 




a 

as 

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ro -4- lAvd t^oo OS d M IN ro -4- >Avd t^oo d> d h pi -^ lovd t^oo OS d w p3 ro 4- 
in in >n m in in invo vovovo^vo^ovovovo r^t^t-~<^t^t~~t^t^ t^oo oo oo oo oo 


00 TfHOO -^M ^^T^o t>.roO\Oro OnvO pi osvO pi CMnp^oo inwoo ^h t>.-<*- 


vO O PI M- t^ PI inoo rovo oo m rovo osw ■'J-t^OPi mt~»o ro moo m rovO 
Oxo^O O O M M M M PI PI PI PI rorororo-^-^'^^intn m\o vO ^ xO c^ t^ t^ 


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« ^VOOO P. ;JVOOO O PI ^vOC« O^Pi^;*;vO^O N ^VOOO O « ;*vO00 O 



308 



FUXCTIONS OF A ONE-DEGREE CURVE. 





N -"l-^O 00 (N ->1-vO 00 M 'i-'O 00 O N ■<*-\0 00 N ^VO 00 (N tJ-vO 00 « 


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00 lOroO t^iAN Ot^rt-H OMD mOOO lOroOOO lOroOOO iDroOOO inroO 





OnvO CO O t^ "^ I 



w ro Lovo 00 O M ro in t^oo O L. . . . . 
OOOnOO>OOOOOOhmhhm 
lO m LT) LO LO^ VD \0 'O ^O vo VO vO "O vO *0 ^O VO VO ^ VO VO VO ^ *0 VO vO vO vO ^ VO 



>o VO t^ t^OO 00 < 

io\o t^oo o o 



On 0\ O O O 



IT) irivo \o f^ t^co 00 O 



OvO (NOO -^OvO NOO -4-OMD MOO -"i-OvO WOO 
Th\0 O w ^ t^ O N 



ro m ■<*■■* rt- ^ 
r^ t^ r^ t^ t^ t^ 



00 Tj-OvO NOO -^O 
rOM3 0\ I 



t^ t^ t^ tv t^ t^ I 



.t^t^t^tvt^tN.t^t^t^t^ 



M '^MD CO O 



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<S Tl-vO 00 O (N -^KO CO 


S S ^^-^ %,'^t>'^%%%t'%^^^^'^%^ 


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t-« rf H OO lO <N OVO C-O 


t^roo c^Ti-Hoo uiroo ^^■<^M00 inw r^^woo 


t^ t^no OO 00 O* O O On 


rOO ON M rj- l^ ON N inOO fONO 00 H '^t^ONN inl^ 

O M M M H (N (N (N rorororOTfrj-Tf^ioinu-. 
l/-)u-)>nlnu^^nlrNu->lOu^tnlnu^lnu^■'^lnu^u-l^i^ln 



lO P) On IT) O On in 
M n -"l-vO OO On M 

IT) in lo IT) in in in 



IN ONin N 00 -"l- 
00 On M rO -*nO C 



I in in in in I 



in PI Onvo m o t^ 

■4-MD 00 o M 
On ON 



. t^co OO 00 00 OO 



g 


r^ r^ t-,00 0000 ONO^ONO m h h w n (n mrorO'i-Tt-Thinin m^o vo \o « 


o 


■>j- in\o t-voo On M 0) Th inso t^oo o> h w ro ■<*- in^o r^oo on m w co ■* io 
„HMMMM<N(N(NP^p^^lPl(NC^roromromr<^rororomT^'^-«-Tl-T^-Tt• 

MNNONCNINDPJC-lWPJNWNNNNNNNNMNNPiNNMOW 


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inM t~.roONO Noo ThO t-^mONinn't^rj-ONO <Noo 'I-h t^mONinwoo -^ o 


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inoo o m\D CO w rONO on m -t^NO onn -j-t^O N int^o m moo rONO oo m ■* 





I ro ■>*• ■>!- -^ ■<^ T)- in 1 



FUNCTIONS OF A ONE-DEGREE CURVE. 



309 



N ■*vO 00 O « ■*vO 00 O 



•^ f) O 00 t^ ui fO t 



O t^^O ■* N O Ov I 



t^ t^ t^ r~. I 



t^ «n r^ w 0^^ ■*« OOOvO -*M OOOvO -"t-i 

\0 t^ C^ M r^i -t-\D 00 O P4 ^omt^o^« C^ -^MD c 
OOO00000i-'-i"i-'i-"0INNN' 



O t^ lO fO t 

. O^ O P) -*-vO ( 



I invD t-^ t^OO O^O^0 w w N MrO-<t-U1 invO t^ t^OO 00 On O 

6^ 6 H 0) m -^ iTi t^oo OS O 



T<-i rj- mo t^oo OS O M ro -*- mo r>-oo o- O 
01 w (N N 0» N N fOrorOfOfOromrorOTj- 



wo N t^ro0s-<l-0"0 

t>6 ' 



OOOnOnOsOsOOOwwwi-iNNC 
C» 01 01 01 01 rororomrororOfOrot 



r^ O ro moo o ro moo 
01 corofOcO'^'^'^-^ 



fO OS -* O SO w 
roo t 



00 O N -^O 00 O 01 -^O 00 O N -^VO 00 I 

cj rorororocOTj--^ii--'i-->j-mmiom mvi 



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O N tOOO I 



I O 00 O •* 0) OS t^ I 



no OS 01 
OsOnOsGOOOm 

- - ' . p» t^ t>. ' 



t^ t-» t^ t-» l-x 



MOOvO rOMCOO rOHOOO mhooo roHOOO rON ost^\r>ronooo -"J-oi o 
m -^o oo o w ro in t-^oo o oj Tj-mtv.os>-i o) -^o oo os w m m t^oo 'o oi TfO 
■<^-<l-■<}•Tj-mmmu^m mvo oo'OOso t^f^t-~t^i^ i^oo oooooooo osososo- 
\0 ^^ ^O ^^ ^^ ^O "^ "^ ^O ^^ O O O O) o o o o o o o o so OOOOsOOO^ 



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o<oow>-'oioimmTfTj-m mo o 


t^ t^OO OOOsONOOHHNNfOr<T4-»n 


o 


t~. t^ t^OO OOOOOOOOOOOOOOOOCO OsOSOsO>OSOSOsOsOsO 
WOlOJNOlNOl^NOJNOlNOlOlOlOlOlPIOlNOimnrOfOnmfOrom 




Q 


GO 01 t^roosmw t-«o)00 tj-qvO n 


t^foosmw r^oioo -<^oso oi t^mosf,. 


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01 -^t^oscj ^r~.o 01 mt^O n moo 


O ro moo M (DO 00 H ■<^o OS M -^o OS 
m m m mo vo o so r^ t^ t^ t>.oo oo oo co 




O N ■*vO 00 01 -^O 00 01 -"i-O 00 
HMHHMNOINOIM 


cororoforo-<l--<i--i*--<l--*mmmm mo 



310 



FUNCTIONS OF A ONE-DEQREE CURVE. 









00 




« *vO00 2 ;2-^oo c; ^vg c« O « ^vo « ^ ^vc 00 JJ^ J«0 00 O 


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ONOO t^vo ^ ro w w o\oo t^vo in tkn N M onco t^vo MDinrhcnM « o 


rn^ ON inoo h -4- t^ On N moo'i-. 4- t^ ro^ 00 m 4- t^ o rOMD o^ N inoo w 
OOOOOO O^O^C>0^0^<>O^O^O^C^C^C3^0^0^0^0^0^0^0^0^C^C^O^O^C^O^O^O^ 


u 


in ro N M ooo <o in rt- N w O ooo i^vo in ^ m f) h o ooo t^^o m -t- m m 


int^OMi m^^oo D Tfvooo cjm- m m t^ omh ro in t^oo N T^vo oo n 
O H H M M M cj n N (N cj w mroromm^^'^'^Tf-'^inininin invo vo 
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 




d 
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ooo^OHMwro^Tf in^ t^ t^oo o 6 h is (r) ^ invo t^oo o> O h (m m •^ m 


N ro in^D (^00 o> O H N m Ti- invo t-^ o w N ro tj- invo t^oo h n ro •* w1 
r^ t^ t^ r^ t^ t^ f^oo oooooooooocooooo o\o\oio>octi0^oo>o O o o 






t^ N t-^ moo moo -^ON'i-OThO ino innvo h\o n t^o t^ moo moo m o< ■* 


1 1 1 1 1 1 1|.|.^^ 1 1 1'll-^^l. 1 1 1 g 1 1 1|, 1 g g|^ 


Z 

S 


C. ^.OOO fj ;JVOOO PJ ^vo« CJ^MjvO^O « ^.OOO O « M;vO00^^ 


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C ^VOOO PJ ^vooo gj M-.0<« fj.^^<« O ^^.OOO £. 5;VO~^ 


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o t^MD f'^H 000 t^-ffN H o\oo t-» in ■* m H o onoo vo m ^ m h o o 
00 o mvo o\ N inoo o mvo o^ w m t^ o m>o o N moo o rr^^ o (s moo h m 

M H H H (N N N mmmmTf-^rt-mmin m^ vo vO t^ t> t^ t^oo oo oo o> 0^ 
OOOOOOOOOOOOOOOOOOOO 00 00 00 CO 00 00 oo 00 00 00 00 oooooooooooooooooooo 


X 


000 t^m'i-N ON t^vD rh P) H ovoo vo -*mM ooovo m-*-mN o o^eo r^ m 


1 E^ ^ ^ ^ ll-Ml-f 1^ ^ ^ ^ ^ Iff fl-l- 1 1 11^1^ IMf 


Q 

o 

d 


vo tN. t^oo ovO O H N mm-* mvo vo t-^oo o\onO h n w m^i-m mvo t^oo oo 


O H p) m -^vD t^oo ov o H (s m Tj- mvo r^oo o h n m ■>*- mvo t^oo 0> o w w 
•<l--*-*-*-*Ti-Tj--<t--*mmmmmmmmm invo vovovovovovovovo t^r~.t^ 
■■Tmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm 




g 
o 

3 


fomOvo M tv.CT t^ moo .i*-o>mo mnvo « «-^ moo mo\.*0 mHvo m t~ 


ii^Kr^iMiiirgiiiiiiiiiiiiiiiiii 


1 


« ^vOCO 2 ;^vO00 O CJ ^vo^ CJ^^VOCO « ^VOOO O « 3;vO«^ 





FUNCTIONS OF A ONE-DEGREE CURVE. 



311 



— 1 


i 


2 


o N ^voeo o cj 2-^0 c« o cj ^vg<« ^^^%%%^t'^%^^^%%S 


Q 


■<*-'*-<l--<J--<*-'*fOfOfOCOfOfOP) N N M N N N N N N N W N MrOfOfOrnM 


O N «/100 M Tj- ts. O rO\0 O IN irioo M -^ t^ rO\0 0^ Pi lAOO H rj- t^ O MvO O 
VD g. g. g.C» Cg(» g^g^g^g^O O O m m m n pj oj cj roc^ro-<^,^^SSlOloln 


inininiouiioinioioioioioioinioioioininmioioiomi/^ioioioininio 


J 


vovovo u^lou^■*.^^^^T^■<J-■^TJ-.^-.J-■.J-Tl-T^T^■*■*TJ-Tt•rJ-^..l-,^•■*lr> 


O P) -^^oo O N -*-«oo o P) "^-vooo o Pi ^^00 O Pi ^vooo o Pi ^^eo o 
Pi Pi Ci P) PI rommroro-*-<i--^->j--^ir)ir)mm ir,\o yo O \o \0 t>.t^t^t>. t^oo 

O^O^O^C^O^O^C^O^O^O^O^O^O^O^O^O^O^O^O^O^O^O^O^O^O^O^a^O^O^O^O^ 




Q 
X 

o 

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in\0 t^oo o> H Pi fo ^vo t-^oo o\ H Pi <v) ^ in t^oo o m pi m ^ in^o oo 


00 0> O 11 Pi Tj- invo f>.00 0\ O H P) -*• irivO t-OO 0\ O m Pl tJ- tno t^CO O O m 




i 


v£) H\0 MVO O ir, O iT) O ino lOO >no^■.i-0^'*•0^'*■0^-*0^ -*00 rOOO f^OO fO 


S5 Si Sv£> vovOvOvOvSvSvOvOvOvSvO \0 vO vO^vO VO vO^vo'vOvO vO vO^vo VO \0 vS \0 




O N -^vo 00 O P) -^VO 00 O Pi -^vO 00 O N -^vO 00 Pi ^vO 00 O Pi -*vO 00 O 


i 


2 

^ 


O « ^VOOO 2 iJ^oo g ?S ^^<S ^^^'%%^^t%'%-^^^%%'B 




O OvOO 00 t^ t^vo min-*rOrOMP)HOOOO OOO 00 00 t-^ t-vO vO vO m lO ■* 


1i 1:1^1: 1 1 1 1.8 § o S> a S, 1 §, S, & § &! n H & a S^ll 9, 1 


1 

^ 


fo Pi M M o 0> o>oo t^ r-.\0 >nio-<*-rofOPi h h o oo ooo oo oo t^ t~. t^vo 


0000000000000000000000000000000000000000 OvO>OvO>CT>OONCTiOONO( 




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m>£) t^OO Ov M Pi f) ■* »0\0 tvOO 0\ O l-i Pi PO •* mvD t^OO Ov M Pi (^ ■<*■ lO 


|||||j|||||||||||||H|||5||SsSj 


§ 

5 


■*0>-*-0-*CT«-*CMOO »O0 »nO too lOO lAO u^O Wh\0 h\0 mvO mvO 
miniommiomirimioiominminioioioioiomirimiDmuiiotniomin 


55 


P< -^vOOO O P< -<t-vO00 O PI -*vO00 O Pi -^vOOO O PI -"J-vO 00 O PI «*vo CO 





312 



FUNCTIONS OF A ONE-DEGREE CURVE. 



ij-vo 00 o w ■*vo < 



vo vo t^oo o o O H Pi ro -"J- T^'m'0 r^ t^oo oi i 



r'l Th \rtKO t^oo O O 



O rOvO O N MD CTi ( 

IT) in Lo lovo vovoi __-_-__ 

irjinmioioiouiiOLomioiomioioioinioinioioioiniomioioioioiom 



t^ t^OO OO 00 



■ lOvO t^OO 0^ O 



ooooooooooooooooooo 



•<*-^ 00 ( 

' cy^ o Q^ i 
o o o > 



■«*- lOVO 00 0< O N (T) -i^^O t^OO O H N 1- lOvO 00 O O N rO -^l-^O t^OO O H N 

mvD r-^00 o\ H P) ro ■* mvo t^ O O w n ro rh mvo oo o O w P) ro -^vo r^oo o^ 
OOOOOi-i>-iwHMHHH(NP)WP4N(N(N(NPirorororr)m(r)fOror<i 
iriioinioinioioininiriininininioininirjioioininuiiriiriinininiriiriio 

■*oo rot^NvO H iDO TJ-o^ fooo P) t>. m in O •>*• CTv moo Pi t^ w ^o O >n o\ -iJ-OO 

PI ■^p~.c^M -^t^ONN -^-vd ON H Ti-yo On h tj-vo 00 H cn\o 00 M rOVD 00 o <n in 
-*■*■*•*■ in in tn mvo no mD vo t^ t-~ r^ i>-oo ooooooo>o>OnOnOOOOwmh 

t^t^t^t^t^t^t^t^t^l^t>.t^t^l^t^t^t^t^t>.t^t^t^t^ t-OO OO OO 00 00 00 00 



O PI -^NO 00 O PJ -^NO 00 O P) -^NO < 



O « ■<)■ NO 00 O PI M-NO 00 O P) 



cj M CJ PI P4 m ro ( 



rorofO-^Tj-TMom in\o no no no t^ t^ t^oo oo on on O 



PI PI ro ro •>*• »o >n\o 

d 

tn 

NP)WNC*P)PJP»P1P) 

mmiotnmtoio 



m m mvo no no tv t>. i^oo ooooo>ONONOOOOHPjpimM-Tf mNO no r^oo oo 

O N "^no 00 O P) ThNO oo O PI ■"♦•no 00 H mint^ONH roint^ONH fOint^ONH 
ooooooooooONONO>ONONgOQOQHHHMMP)p)PiP)pirororor<imTt- 
OnOnOnOnOnOnOnOnOnOnOOOOOOOOOOOOGOOOOOOOO 

MMHMMHMMHMP)P1PJPIP4P1NNP»«P)P1P)NC»C»C»NP)NC« 

00 ON O H P) -ij- mNO (^00 O 1-1 Pi fl -^nO t^OO ON O PI O ■* mNO oo on O m pi ■»*• 
H PI ■<*- inNO r^oo On O H rn 4- inNO t^oo on o h m -4- inNO t^oo on d pi co ■4- in 

t^ t^ t^ t^ t^ f^ r^ t^OO 0000000000000000 OnOnOnOnOnOnOnOnOnQ o o O O 

•«^•^■^•*■*■<»■■<l■TJ-■«*■•^j-^■^■^■^•^■«J• «j--^-^'*Th-<*--^-*Th'*minmmin 

moo PI r^PiNO MNO H mo mON'«*-ON coco rooo pi toNvO mvo OmOmo^•<^ 

00 o m "100 o m moo o mmr^g pi mt^o m mt>.d w mt^o n mt^ONcJ 
NO t^^ t^ t^^ t^oo oooooooNONONONOOOOMHHHpiNNp»mmrom(r>if 
nonononononononononononond r^t^t^t^p^tvi>.t^t>.h^r^t^h»t^r^txt>.t>. 



O N •*vO 00 O 



FUNCTIOX"^ oy A OXE DECREE CURVE. 



313 



O « -^-VO 00 O N -^^ I 



O f) ■*« 00 O *• ■*vO00 O 



CO •<*-\0 00 O N 



invO 00 O N ■<^lr)t^O^M romt^OH fO inoo O N -^vo 00 O fO lO t^ cy> M rovO 

o H (Ti^o 00 O N -^^ c^ H CO in t^ Q n -^^o o m m id t^ o n -^vo oo m ro m 
\0 t^ t^ r^ t^oo OOOOOOOOOOCyiO^OOOOOHMMMNNNPiNrororo 
MMMHiHWHi-li-lwi-lMMM(NMPINNN(NN<NNMP)(NWNNN 



g 


\O00 OH <N -^lOfs-OO H rO Tj-\0 t>N On O « M mvO 00 On h W •* in txOO N 


o 

d 


ro Ti- in t^oo o o H N -"i- inNO t^oo On O w fn •* rnvo t^oo o h m ro ■* m r>oo 




Q 


in ON ro (^ H in 'i-co M no O -^oo o^o h inONrot»H inONfntv.NNO O ■* oo 




00 m inoo O romt^O w int^ONM -"j-t^ONH -*no On h roNO oo h rONO oo O 
00 TO^ro C» m" OnOnOnONOnOONOnOnONOnOnOnSSSSonOOOOnSonOnOn 


Z 

S 


W -"l-vO OO O <N ■'tvO 00 N ^VO 00 O N "^vO 00 O N ■♦NO CO O N ■*nO 00 Q 

„ H M H H w N (N o p) ro M roTo or* ^ Tj- ,^ ,^ m m m m mNi 



< 


ro ■* in t^oo ON O P) ro tj-no t^oo on h w ■* in t^oo O m ro tj-no oo on h n ■* m 




en 

X 


00 On M N -^c inNO 00 On H N m inNO oo on m m in f^oo o w ■♦no r^ on h m m 





m •<*-no t^. On o h ro -"J-no 1^00 o H m •♦ in t^oo o h w ■♦ in t^oo on m n ■♦no 

0> O H IN ro inNO t>.oo on O H m -♦ inNO r^oo On h N ro Th invo t^^oo O h N ro 
ro-^-^-^-^-^-^-^-^Tj-inininininininin inNO nOnonononOnonO t^t^^^t^ 
inintninininininininininininininininininininininininininininin 

00 W vO H in On rooo MvO O ■♦ONrot^.HNO O ■♦00 fO t^ h m O ■♦00 N nO m m 

moo O romt^O N mt^o in 4-t^O>iN -i-t^ONH •♦no On h ■♦no oo h rovo 00 
w H p) N M (N rorororo^^^^^^-*-*-minm mNO no no no t>. t^ t^» t^oo 00 00 oo 
oooooooooooooooooooooooooooooocooooooooooooooooooooooooooooooo 
^>.^^t^t^txt■^t^t^^>^^^^.t^t■~t^t>.r^t^t^^^r>.t■^r^t^t■~t>.t>.l^t^^».^»r>i 



314 



FUNCTIONS OF A ONE-DEGREE CURVE. 





i 




1 

« *vO00 O JJ 2-vOOO gj ^vO<« O « ;3;vO<« O N^vOOO « jjjvoco^ 


< 


00 H Tt- t^ COvO OMH ■* t^ O rOvO 0> N lOOO N UIOO H -^OO M * t^ -*• t^ 
O ;?;P;ii^O j>J^O ^^O^covo Ojr>^ d gjvo j,vO O rovo O rovo 




vO 0> (N lOOO M -^t^o tpt^o f^f^o f^t^o fpt^-o <p«7-o fpt^-o rpt^o ■>*■ 

ro irioo 6 (N lO t^ t> N -^-vO 0^ « rovO 00 c5 roiOt^O N -^t^ON" -i-^ 00 h rn 
N N CM (N 0) N c?c?c?c?c?c?c?c?c?c?c?c?P) c?c?M c?c?c?(N c^c^eTerN 


Q 

o 


M r0l0^«0^0 N -^O 00 (> m fO in t-.0O O N 'J-^ t-.00 N -f^f^CTMH fO 

ro -5- Lovo t^od d M (N rn ri- in t^oo c> w ro ^ invd t^oo O w (N ro •* in t^oo 
•<l--<j--'*--*il--<l-minioinininmin in>£) vo\Ovox>vD^'0 t^r^t^r^t-^f^t^t^ 




Q 

g 

5 


■*oo H inoiNvo o -"i-t-^M looo f) vo o m t^ H '<^oo w m a. rovo -*-oo m m 


p) •* t^ O w i*-^ O H rovo 00 roint^o roint^ON -^^O 0^ m -<t-vC) oo w ro 
.gcg I" J^i lilcg eg cl'i'l 112 cS eg eg eg eg S III eg eg <g eg J eS 


2 


C ^vOOO 2 2-^00 O CJ ^vO<« O CJ^;*vO.» ^^^00 P, ;5;^00^ 


00 

cc 


2* 


c ^^00 o cj ^^e« o c. ^^,» o « ;5;voe« ^^voco O « jvoe* O 




mvo 00 H ro inoo O <M in r> c^ N •«^vO Onm •^^t>.o^M -^t-t-^O N "lOO ro inoo 


inSSininlnlnin In'Sj^'Si Jo lo in'Si^'^ Si Si invo vovovSvOvOyDvOvOvO 


u 

X 

W 

Q 

o 


vo 00 M ro\0 00 M m^ 00 M ■.l-t-.OM<l int^(>M inoo ep^ <?"^ "* ^ 9 rp^ 

in t^ O N •^'O O H ro inoo O n •<*• t^ C3^ w fovo oo O ro in r^ O w -^^O o w ro 
rorO'>j--^T)--4--*ininin mvO vo vO vo vO r^ I^ i^ t^oo oooooooo O^O^CJ^C^0 

S ?! S S ?! S S S ?! S S S S S S S S S S S S S S S S S S S S ???? 


p) ■* in t-^oo O N ro in\0 oo O m ro -^O oo ov w P> ->J-vO t^ 0> w -* m P-.CX3 


"§ 8" 2 S 2 M M "S M 'S 2" S c3 N N N "S n" ^-^ fo ?o ?o m"^ P^'^ ^ §- !3- ? 

vOvOvO^vOvOvOvOvOvOvOvO^vOvOvO^OvOvOvOvOvOvOvOvOvOvOvOvO^vO 


§ 

X 

o 


<» MvO fOr<.H »ne3>rOt-vH •♦oo N vO O ■♦oo m mONrot-^w inO\NvO o ■<»■ 


7960 
7963 
7965 
7968 
7970 
7972 

7975 
7977 
7979 
7982 
7984 
7987 
7989 
7991 
7994 
7996 

7999 
8001 
8003 
8006 
8008 
8010 
8013 
8015 
8018 
8020 
8022 
8025 
8027 
8030 
8032 




« *vOoo « 2-vOoo jj jj-vo^ « *vow 5.« 5^e« £. 3;vOe»^ 

• 





FUNCTIONS OF A ONE-DEGRSE CUR VI 



315 



■*0 00 O C" ■^■•D I 



, rO r^ w "1 On I 



\0 O -^Ovrot-^NvO o •<^o^ 



iritoioioiO'-Au^io^^i^' 



M ^ I- o C) rri rn 
On O 0> O C> O^ O 
to IT) lO u^ ir> lo I 



■ ON r<5 r^ N y 
■^ r-> On " ^NO On • 



I lO lONO NO NO NO t^ t>* 

Cni(N(NIN(NMNCS 



00 O « -*nO t^ On h ro id I 
m lONO t--00 On O >'N' ro tT I 



■*NO CO O ^ -^NO 

CI m Ti NO t^oo qn 
t^ t-- tv 1^ t^ ^ 



Onn uiOnN ioonN mM IN) u~00 P! lAOO M inoo M -*oo H -!i- i-^ m r^ O r^NO 



:i r>. O N -^ t^ On w T)-V 
00 OnOnO>OnOnO O ' 



ro looo o P) lo t-- On 



rot^O -"l-t^O -"l-t^ 

O fONO O rONO O <^NO . _ . 

m m ro ■* ■* -^ lO >') triNO NO NO t~^ t-^ t^OO 00 00 On On On O O 

t>.t^t^t-»f~t^t^t-^t^t^t^t^r^t^t^t^t^t-^t^t^ t^oo 00 I 



CNi NO O ro t^ H I 
t^ O ■* 1-^ O T)- I 



ON mvo O -J- t^ I 



1 ON o^nO O -^oO (N NO O -^QO 



I NO O r^ On ro 
I O n Lo t^ O 



u->oo O N Lor-0 N Tj-t>.ONH -"J-nO On m conO 00 r- m 

r^ t^OO OOOOOOONONONONONOOOOHiiMiHNtNltNtNr'inmm-fl- 

r^^rom-nr'^mc-oo^mror<^T(■■<J-■^Tt--*■*'^-!^T^•^J-■^r^Tl-■^Tt-■<t•» 

NN<NINCIN<NWM<NNPlCNNNNNO(NMNN(NIN<NNNC» 



rf> joNO 00 O N 



IT) P» On w fO lONO 00 O 



On O N -^NO I 



) CO 00 GO eg po •» OO .ON On ON ON On On On On ON O 



VONONOvOVOnOVONOVONONOnononononO 







tv t^ t^ r^ 



noOn'n-ino O ro t^ O 



•*00 M ICC 



o 


S^'S 


C 


j i/N f^ On C 


1 M 


u-s^ 


^^'^%'^ 


^ '-* 


00 


?, 


Jn 


s 


in 


^^ 


.^^Ng^ 


tv tC 


00 00 CO 00 00 00 00 00 00 00 00 00 CO 00 00 00 00 00 00 


□0 


00 


00 


00 


00 


00 00 


00 00 


00 


00 00 



l-NO CO O M Tf-NO 00 O 



316 



FUJSrCriJ.V^ Of A OyE-DSGfliEE CURVE. 



■*iO 00 O M 



(TlOO Tl- ON Tj- O "^ I 



i ^o yo 



\0 vo ^ ^ *0 *0 ' 



^OOVO^O^OVOVOVO^ 



ro 0\ "il- O vo H t^ rooo ■* O 'O 



Tfvo Cf. ^ -^ I 



-^ f^ ON CM LO I 



Q 
O 
Q 


t^ 0\ N ■^'■O 00 N ^'O 00 H m m t-^ C3N M <r)\o 00 N •*"0 On 1- ro in i^ On N 


inNO 00 On M m Th inNO t^ o^ ,« o m inNO tvoo o « <N m -^no r^oo On N 

OOOOOOOOOnOnONOnOnO\OnOnOOOOOOOOi-imhhmi-i-«>-N(ni 


S 




0* 

o 

K 


00 M -^ t^ rONO CO -H rONO On N moO w ■* t^ Q ro irioo h 'I- t» On N moO H ro 


f» in t^ On M 'tNO CO - romi>~o <n T^t^o^- t)-no oo -^ in t^ on n -"^-no on >- 
H M M r- (N IN N (N (^rnrom-*Tj-->^rr-^Loinm mNo no no no no t^ r^ t>. t>.oo 
mro mrorn-^ro'or^'nroro(^mroromroo-)roir,mrororofOfor^r<- ro ro 
00CO00N/00O000000000O00COOO0O00000O0O000O000O0O00O3 00000000O000 



^nO 00 O in ThNO 00 o 



I ro ^ -a- -^ Tt 



NO O m O 'i- On -*c 



mt-~0 rj-fvQ ^r^M Tt-oo M inoo w m On m no Jn mNO O r^, t^ o -^ t-% h •<^oo 
m CO -^ -i- "^ in in fr,NO NO NO t^ t^ t^oo oocoONONONQOf-t^MOJOJwroforo 
O^OnOnOnOnOnOnOnOnOnOnOnOnOnOnOnO'OnOnOnQ O O O O O O O O 



in in in in lo in 



in in in '-n in 



-,no 00 H roNO 00 w rONO ( 
Ninmininininininin 



O NO M t^ N 

■^NO On I- ■<*• 



Q 
O 


NO 00 (N -^NO CO M -Tl-NO 00 N T(-NO 00 


(Ni m, t^ONH roini^ONM minc^ 


O- N ro ■«■ invo CO o M IN Tj- invo t^oo 
■^ininir, inuninm inNO no no no no no no no r-. 


- (N ro ^NO t^OO O- M "^ •* in 

1-v. 1^ 1^ t^ t-^ t^ 1^ rvoo 00 00 00 00 


S 




Q 


NO a^ o NO ON 0) moo (N moo M T^ (^ -^.^ on 


IN moo - ■* t-^ roNO ON M inoo 


O 
X 

u 


rr inOO O ir rv on N -*no On 1- r<-)KO CO O •nj 

rl- -.^ -«- in in in m mNO no no no f^ t^ i^ t-^oo oo 

(NMNNPiOClINIMNINNrNlNNClINN 

oooooooooooooooooocooooooocooooooooo 


ir, 1^ On (N) -^NO On M m moo f> 
cooooo OnOnOnOnQ - ►- 

oooooocoooocoooo oo'oo'oo oo co 




CN ^NO 00 N -^NO 00 <N -l-NT CO « -1-NC 00 O P) ^no 00 W -^^O OO 



FUNCTIONS OF A ONE-DEGREE CURVE. 



31 





a 


2 


O P) ■*« 00 P) -^^ 00 N ^vO 00 (N THO oo P) ■<^vO 00 P« -*-vO 00 
„„M«MP)P)M(NPimmrornrnT»-TrTj--<r-<i-mmmm mxJ 




N ONmPioo mpioo mpioo mp4 ovo P) a^o en O r^'i-Moo mcj o p-tI-hoo 


mvO ->-t-^M TOO Pi^ o-'t-^O M-20 -1 mON rno -S- t^ i-- m o.pt vo rr 
m <^^ « o i^ t-^ 1^00 ooixioaNOOO«-'-'-pjp<'-o<v^m^ tj-.^ m irtvo^ 
PI PJ PI ->- PI PI p, PI PI .M P) PI PI rn m ro .-n n m ,0 ro r^. r^ r'.. r'-, m S m ro rri ro 
vOvOvOO\OMD\OVj.wVD^V^V.^vO«\OvO»0^vO>0\0'OVOnOOVO\0 ^ AO VO 




m P) ovo rnC f--^-;oo mp) '^■^'■^ O-vo m^^oo^o mwoovo rowoo^o ■*- 
w ■^^^ o N m i-^ ri ir,co " 'j-^o o PI rf p.^ Q ee; inoo " Tj-'O OP) m r^ o' en 
m m m m^ ^ \o t-~ t- c^ p-oo oocjcooooo c c - ^ - " p< pi f? J^ Jv^ 
J^5:-5:-J^^^^5^^S~«^™^!^5^5^^°°°°°°°°«' oooooooooo 00 00 

p)P)PlPtPIPlP)PIP)MP<PlP)P4NP4PJP)Pipipjpipipio.MPIP)piwPI 


d 

OS 



d 


O « fovo 00 q ro in i-^ o p) tj-^o o rn ro^ oo m moo o P) m p^ o p* -<t\o cy^ 

"S-A vS VO vD i'S VO 5 vg^ J^ fi K ?:^ P:^ gltS cS ^■^'g ^^^^i^x^ 

OOOOOOCOCOOOOOWOCOOOOKCOOCOO TO MOT §5 S S OT OT So S§ TO TO TO TO OT <g^ 




d 
o 

u 


Pi mr^o P) ■4-r-.op) ^t--op) Tj-t^O" t)-vo o -< f-.vo to o m mTO o p) Th 


O »- r<~.vO TO PI •'t-'^OM re.KO CO Pi mP^On ^i-vo TO O m in tv o P) -^vo 
T^ m m, in invo vO vO vo \0 t^ r^ r- P^TO TOTOTOTOOOO0-00000--W 
•<^Tt-T^-•<t•,«-T^-Tr-.^T)-^T^^T^-^-T^T^Tl-Tl-T^--^.^Tt-Tt-^nmi'^mlnmmln 

TOTOTOTOTOTOOOOOTOTOTOTOTOTOTOOOTOCCCOTOTOTOOOOOTOTOTOTOTOTOOO 


2 


PI -"J-VO TO PJ ThvC TO PI ThvC TO P) -?VCTO O PI ^MD TO P) ^VO TO O 
M w rn M M Pi P) P) PI PI rocor^. o-l,CT^T^T)--r-^mmmm u.\o 


e* 

0» 


2 


P) ^VO TO O P) ^VO TO O P. ^vo TO P)^ ^VO TO « ^VO TO Pl^ J;^0 00 ^ 


d 


t-,mosThOvo P)TO Tt-Qvq P)00 mM r~-rnO\mpiTO •<*•«- t~..rn0'O en o-O P) 
■*TO - inOPivorrvrnt^o -*r-M mco N m o\ mvo ■* t^ - mTO pj m o m 


VOVOVOVOVOVOVOVOVOVOVCVOVOVOVOVCVOVO-OVOVOVOVOVOVOVOVOVOVOVOVO 


u 
u 

x' 


TO ^ t^ rn OVO PiTOvo MTO ^m f^^Q p^rno t--PO0 p^tJ-q p^tI-mto m 


M ^t^ovPi ^P-O P) mTO rovo TO m Tfvo o\ Pi ■* P- P) 'nco mvo TO m 

VD'CvOVOvOvovOVOVDvOVD l^r-r^P^t^I-~P^(^t-- P~.t^I^I-^|-^t-^p-,t^l-^t^p^ 

C^N?lNNP)NNP)P4P)NPJlNPIP)P)P)PlPJP)PIPlP)P)P)P)PiPIPlM 


d 
O 

d 


p) Thvo 00 ro in p^ o M -*\0 to m m r-. o P) ^t-vo TO m po m t~. o pj rhvo o 

Pt p<^ ■* m i-^TO ov M po ■* mvo TO o - P) -* mvo t^ o o « P) ■* mvo p^to 
(M P)NP)P) PI PI mmmr'^mmrnro^^-rTi-Tt-Tf-Ti-Thminmi'-mmmm 
totototooototototototototoottotototototototototototooototototo 




d 
o 

K 

u 


cnvo OsM ThPvQ N mp^O en mTO m rovo on m Tfvo onp) r^p^oP) 'J-tvO pi 

m' -n inTO P) in P^ Ov M -j-vo oo m m t-~ o\ P) ■+vo TO - cnmtvO c« -^f^o 
TOCOTOoo osOOOOO """MM p) P) p, PI en e^ en en ^ ^ -^ ^ ^ 

TO TO TO TO TO TO 00 0000 TOTOTOOOOOOOTOTODOOOTOTOTOTOOOOOTOOOTOTOOOTO 


2 


C< 'l-vo TOON ^vc TO P) -*-vO TO O PI ^vo TO Pi — VO TO Q P) T^vo TO 

r- M H iH M p) P) M P) P) mr^. mrom'»--«--3-^^inmmm mvo 





318 



FUNCTIONS OF A ONE-DEGREE CURVE. 



OO O ■* N 
M ir, o -o 

o o c^ o 






VO vc ^ ^ *0 vO VO ^0^4J ^w^^ VO vO ^ 



m r->cq o O f^^ 



ro N ^ 0:X) t^vO M- I 



N -I O OOO t^^ in ■* m N 



t^ O m<3 00 - -"l- t^ O ^I'O oo M -f (> O roMO O 0) •* c^ o ro>0 O P) moo m 
1-1 N (N N N r<iroro^-<t-i-l-u^Ln uivo -o vo ^o c^ t^ t^oo oo co oo O O O O 



OS(N rJ-t^OsC) tl-t^O(N ■* 



•^ IC t>.00 o o 



OS On On O 0^ O 



O^O^OsO^OvO" D^ O^ O^ < 



1^ On -I CO 1 



ON ►- ■* C 00 O IN lO t^ O- « rovo OOOM ■^t^ON" rOtDMOCN Ti-\0 On 
COOnOnOnOO ^000'-''-'"►-'«(NPlCNlC'^<-or<^r<-)ro■>l-■»■•l-■<J•T^ 
in LO 'n IT) iriNO ONO OnONONOnCnO'OVOnONONOnONOnOVO'O^NOnOnO 

oooooooooojooooooooooooooooooooooooooooooojooooocooooooo 



■>l-NO 00 O f) -^-Oou O 



m O 1^ 
. - in 00 



O ON ro r^ I 



NO NO 

m rn m m rn -n ro -o -n 

NONOm_ ^nOnOOOnO 



o> r^ -"f ( 

-TOO 



O - H 

NO NO 



On t~« in N 

On CO t^ « 



■+ -t- Tj- Th Tf- -^ 



NO OnonOnOnOnonO 



■<*- M On f-~ in 

r'-iNO oo M -f- 



inOO M rJ-NO On 



OnOO vO -"J- ( 



Ooo i-^inrri- 000 rNin 

' N ■* c^ O rONO On ^ 'i- t^ 
OnConOOOO-'- 
000000 OnonOnQnOnOnOn 



0> w -«->0 00 M <r)NO 00 ro inoo O mini-~0 Pi int->0 N mr^o N -1-i-^Oni 



rn-^inr^oo onO n ro-^in i^oo on o pj 

OnOnOnOnOnOnOnC^OnOnOnOnOnOnOnOn 



-r in t-^oo On o ►" 
OnO-O OnOno^OnOn 



■* t-« On w Tj-VO 00 1 

NO 00 6 ro in t^ (> 1 
in in in in in in in I 



in r-~ On N -^t-NO 00 O ro in t^ On - 
NOoo O ^ini^o<N -^NO 00 O rn 
m in in in fn in in in in I 

OOCO>J0OCO0OCO0O0O0 



-l-NO 00 
1-NO 00 O 



o p) -"i-NO 00 o 



■*VD 00 O N -^NO ( 



FUNCTIONS OF A ONE-DEGREj: CURVE. 



319 



Si 




M -^NO 00 M -fNO 00 <N ■<^NO 00 P) Tj-^O 00 0) Tj-O CO O N -fO CO o 

M H « M « p) p) pj P) M m(-ominrr,Tt.,r^^,MDiovom iON5 


Q 
< 


q 0< OnOO 00 CO f-- t^ t^NO NONONONCVONONONOVCINONONONONONONDNO (-.(-^ t^OO 

o- P) NO d -^oo" pi ^ ^oQ- P) 43 d -^00- pi NO d 'i-od pi NO d ■'i-oo pi o" d •'i-oo- 
0"'-iP)P»P)ro<~0-iJ--<J-^Lr) u-)NO NO O r^ c^oo ooocjOnonOOO-i-pipipj 
r.Ni>.t-~rvt^i-^r-^fvt-^t^t^t^t-^t-^rvt^t-^t-^t--,t-^r^i^ t-^co co ^^ oo oo oo oo 00 

NOOnONOnONONOnOnOnOnOOnOnOnONOnOnOnOnONOnOnOnOnONOONOnwNONO 




OvOnOnOnOnOnO " M w PI P) ror'-. ■*T^u-, inNO t^ t^oo On m m p) m 


f) looo w •* rv, w Tf r-, rONO on n looo w -^ t~. rONO on n inoo PI looo w ■* 
OnOnonO 1-1 H M P4 p) PI M ror^^ro•<J-■<^■>i-LnlDm iono nO no t-^ t-^ r^oo 00 


rococororococ^MrO(^rorotr)?oroM(T>?)i^c^P;t^romfriP^(vi(r)m 


Q 

o 

d 


t-^ ON PI lo t-v en looo M T^No on n '^ c^ o ro moo h ^^no on pa m t^ o <r>\D oo 


00 on - PI ro lONO t~>x) M PI m iono tv On w p< •* lovo tN on w ij •* mNO 

8 8 g i 8 S g 8 i 8 i 8 g S S i a S ^S 8 a sTcSHs g 8 8 


q' 
o 

X 

u 


fl ..^NO t^O>>-i PI .*lOt-»O>0 N i-O lONO 00 On H rO ■* lO t~00 O M fO -"j-NO t^ On 


ooooooocoooooooooooooooooooooooooooooooooooooooooooooooooooooc 


2 

s 


O N -*NO 00 N TfNOOO P< ThNO 00 PI Tf-NO 00 PI T^NO 00 o pi -^no oo 

M H 1-1 M M PI M PI PI PI romrocN-)ro^Tf--*-^^Loiomm irjNO 


o* 


2 

s 


PI -"l-NO 00 PI -"l-NO 00 PI -^NO 00 N -"l-NO 00 IN -*NO 00 PI -i^VO 00 




NO lO -<t PI H O ONOO t-vNO lO •<^ rO PJ M ONOO t-^ tvNO lO-^-^rOMPlHMPlO 

M m ON rp r^ i-i Tj-od pi NO d 'i-oo pi no" ro t^ h m d- ro t^ - in ON fo t^ H m On 
OnOnonO «"" N " romro.*-!)-mm iono no no t-^ t^OO 00 00 On on o 

nOnOOnononONOnOnOnovOnonononOnOnonononOnononOnonononononOnOno 


u 
Id 

X 


ONOO 00 t-^NO NOiom->*-mmr0Pl pi m m h o O^o^OnOnonOnOnOnOn 


S"-S S" 2 l^'S ;:J :?- !^ 2n 2'9^ ^ 2. 12°2. " ^ f-- o pono ooM^^.omooNP^ 


Q 

o 

Q 


00 rnNO 00 h pono On >- ^nO Onpi Tj-t^Q n int>.o ro moo m rovo on w ^ t-^ 
PI ro Tl- m t-^oo On PI '1 -^ m i-^oo on ■- pi ro ^nd t^oo o> i-i pi ro -^I-nd r>,oo 

&S;S;S;S;S;S;S;'S>^'§N^'^'^^^S^S;S;s;S^g^g5§ 8 8 8 8 8 8 




Q 

O 

X 

u 


PI fliot-ONH ro-^NOoo PI romt-^ONQ pi -^no p^ On "-i pi ■^^NO oo Ov h ro 


ON iH ro m f^ o> P< -^fNO 00 'O m t-^ On - roNO oo pi -^no o-m romr-^O'Pi ■* 

nJn^n^n^nS^^s s s s nS-.^n5-,^n^s s s ^ s^.^^s ^ a R ^. a ?. f; 

oooooooooooooooooooooooocooooooooooooooooooooooooooooooooooooo 


2 


PI -*-NO 00 O PI ^NO 00 O PI -1-ND 00 PI ^NO 00 PI -^-NO 00 N ^NO 00 

M M w M M M PI p) PI <N rorO(ororoii-TrTi-^.i*.mmmm mvo 



320 



FUNcrio.Y.i o:r a ON!L-Dj:r}R::E cup.rs. 



o* 
H 

© 
H 




N ^^00 N Ti-VO 00 N TTOCO N .^VO 00 (N ^^ 00 C4 -rfOa^ 

„ „ M H « (N (N IN CI w CO .-o .■■ CO CO .a- Tf rr T^ Tj- .o in m U-, m^ 


u 
w 


N CO Tfvo t-^oo HI N ttO t^o^O N comr-~ONO n -"J-vo oo n 'i-vo oo 

M in o^ CO t-. M lo -^oo Mvo -^ONcot^M mo- .<^oo mo q mo^coi^^-^o 
in lo invo vo t^t^oocooo o>onO o o ~ --i ct m c< coco-^-^min ir.vo vo r^ t^ 
aNaNONa-c^oo-1-o-c^aNaNOOoooooooooOGOooooc 


coint-^ONH coint.va\H coint-^ONM '^'O oo co moo co moo m tv Q m 

oo" H .^rr^w n-t-^o coi-^0 co\o Os coiO o>c>ivo o^N mo^N moo N moo oi m 
t^oo ooooONONONO - " " H.C4 CM 0) cococor^^Tt-mm m^o vo vo i^ f- 

CM M IN Ot n 01 CN CO ro CO CO CO CO CO CO CO CO ^ CO CO CO CO CO CO cc, CO CO CO CO CO CO 
CO CO CO CO CO CO CO CO CO CO O 1 CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO 


Q 
K 

o 

c 

s 


covo 00 H ■* r- covo o> M Th t-^ c covo O c^ m.M h t^.^© Ov n moo ►- ^ tv o 


m^o t^ C3^ H CO TT mvo co o o c* co ^j- lo i-^oo cj- ~ (N co ■♦^o t-^oo o p-i pi -"r 


Q 

K 
O 

a 
u 


00 o CJ CO 'i- mvD 00 o^ w M 'o Tf- m t^oo o o m n co ■■i- m'O i-^co o o m 

IN -^t^O" comt~-OH T^vo oo ci -^O 00 ro in t^ o^ - co m J.^ (7- m -t^ 
^^Ti- ^mm m m mg^o^vr^ t^^^^ t^^ oo co OT oo o cy o o 0^ 

oooocooocoSooooojoooocoooocoooocooooooooooooocoooo. ooooo&cooo 


2* 


N -^.^ 00 N Tl-VO oo (N '^vo 00 (N Tj-VO 00 M ^O 00 N -<»-VC CO 


H 


2 


O N Tt-VO 00 N -l-OOO N ^^OOO N t)-\0 00 CX tJ-O 00 IN -^VO 00 




000000C3^O^00HH<NMCO'^-■* mvO t--00 00 O m N CO -* mvO tvOO O 


00 NO -^ocot^'-' movcot^H mocot-^- mo -^oo n vo o -^^oo p) >o m 

1 g s § s ss s i 1 i s s i i 1 f s ll^l^vg^ J-I^IJ^HHI^ 


o 
w 
en 


m ■* mo t-^oo o w N rt- invo t-^ o O m co -* -j^ t-»!» o n r'! in r-.oo o pi co 
ri- tv, o coo r^ ci O c> PI moo m -*. f- m tj- t-^ o coo On coO O pi ""OO pi moo 
cocococococococococococococococococococc, cococ'-;cocococncocococo 


o 

p 


00 H '^ t^ On N moo M roo On P) m f^ coo o M .>*• t^ coo O w -^ t-^ en 


o oo' <> 6 M CO 4 m t^oo t> o' pi " ^o f--oo c> h n co mo tv.oo 6 " N M- m 


Q 
K 
O 
X 


ON w m -i-o t^ o- C3N CO Ti- m f^-oo on h pi co mo r-^oo o h n co ■♦o t-oo 


00 " CO in t^ CN M coo 00 d N -^o 00 6 comr>(3N" co mod 6 P) 4o o<j o n 

t^oocooocxjoo oonoonooooom---mc;^i '^ ^sps;:^:;:^;:^^ 

t^t-^r-vi^i^t-^r-t^t--. t^oD oooooooooooooooooooooooooooooooococooooo 

00000000030000000000000000000000000000000000000000000000000000 


z 

s 


N ^OOO N ^OOO N ^NO eg N ;^0 <^ N ^JO 00 N IJJVO CO ^ 



FUXCTIJXJ Oir A ONE-DEGREE CURVE. 



331 



O N -+^ 00 



nvo 






O rovo O 'tt^i-i ioor<1t^> 

M .00 -rO- rr. t^C4\0 O lOO^- 
_____ .. pj ^ , 

(^ ro ro 



,\0 VO t^ t~«00 00 O On O O O - " 
CN (N (N CI M N M N ro -1 'O ro 



IT) o- c^ O ro -a-00 

1-00 ■- -:t-CO 



r<1 (^ M ^ O 



ro m CO CO cc; CO ( 






O- 0) moo -< ■* r-~ M ^ tx o 
iVO CO t> O CJ CO -t\o t^oo o 



O M u-iOO " lAOO 
CO LOVO t^ O C — 



■* l^ O co^O o m 



. o- o o ■ 



tvoo 000 

CO O CI -1- 



"i^o t^ t^oo o o\ o o ^-^ 



O O On ON On 3n I 



. r^oo o3 00 CO ( 



■ ON ON On ON O O O 



O O O O O C O 



W N ?v C. 



OnOnCJnwnONCT^GnOnGnOnCnOnOnOnOn 



CO ro CO to CO 



O M ■*"0 00 O N ■^'O 00 O N 



^NO 00 O N -*-\0 00 O 



\0 O uo O- cooo 



r-. o coNO o- I 



-*- r». c^vo 



t^ t^ r>. t-^ t^ 1 



■ ovo NO NO r^ t~.oo c 



10 t^ O coo On N 10 On N u^OO 



-.J-OO M lAOO 



t^ r-^OO 00 00 On On On O ( 
rorococococococo-^-' 
COCOCOCOCOCOCOCOCO( 



I ro CO -* -^ ^ 10 in LT.No vo NO I 
T^-T^^Tt-T»■-:)--^T^-rl-^■^. 
I ro CO en ro cT) CO ro CO CO CO CO ( 



co\o o\ M tnoo -■ ^ r-. ro\o on n «oco m -<»- i>, rovo on n 


inoo " ^ t^ 


•^ invo t^ ON w rn ^ -n t^co On CNi ro -*-vo t^oo - rj co m 


NO r>- On " ro 

uo to inNO NO NO 


N(N«MNNC)NCl(N«(NMNC4f)(NCNN(N«C<0iM01 


N « CI <N M W 



i-i N ro -*• inNO t^oo ON O O ►^ N 
-rNO CO O N lo r^ On K 



inNO vo 1^00 ONOMt-Nco-^iT) irjNO t^ 



__ . . _ . ON >- ro ir, 1^ O r; T)-NO 00 O N ^NO 00 

_ Q M t-i ^ — »- rj CI cj PI CI CO ro ro rr I-'- -^ -q- -^ -^ m lo »o lo uovo vo nC vo *0 
OnOnOnOOnOnOnOOnOnOnOnOnOnOnOnOnOOnOOnOnOnOnOnOnOnOnOnOnO 

COOO ^. ooooooooooooooocoooooooocoooooooooooooooooooooooooooooco 



CI «)-vo 00 o 



322 



FUNCTIONS OF A ONE-DEGREE CURVE. 



■z 


n 




■<J-NO 00 


o 


f) -^-VO 00 


n 




T*-\0 00 


o 


(N -^VO 00 


N 


"iJ-VO 00 





« ■"hvooo O 


S 


















. 


















d 


ID 





in Cm 





IT) W 


VO 


„ 


VO 


N 


r^ moo 


m 


ON "i- VO M 


t^ 


rDOO 


^ VO N 00 -+0 


r^ 


(N 


O H lO 


Tj- On rO 


^ 


fN) 


^ 


H VO 


in 


On ■* ON rrjOO 


(N 


^ 


H 


VO 


H 


in o ■* ON-* 












H 




M M rorofO-^-^io mvo vo 


r^ 




<i 


































t^ 


t^ 




t>. 


f- t^ t^ 




l-^ 
















i 


VO 




f-^ ro ON 




ONO 




00 


■* 


VO ro ON tn 


H (V. T^ VO 


ro ONVO 




ON 


in (s ON in f) 


in 


N 


VO 




R 




„ 




(N 


inONWvo 


mt-^M ^ 




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in 


































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en 






mmmmcn 


Q 

o 


00 




■+00 M 


^ 


t-~ H 


-* 


t--.0 


■* 


t^O fO 


t^ ro 1^ rovo 





mvo 





mvo On mvo 


„ 


ro -* lo t-^oo 


ON w 


M 


m 


invo 


l>~ ON 


1^ 


f' ■* in 1-^00 


ON 


„ 


IN 


m mvo t>.co M 










•* 












Lo mvo VO VO VO VO VO VO VO 


t^ 






r^ 




Q 










































































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. 


CO 


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mmm.. 




H 


r<1 










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•-I 












ci 




■+ 


•* 




0^0 — - ^^ 


a^^SN 


OnOnOnOnOnUnOnOnOnOnOnO O-OnOnOnQnOnOnOnOnOnOnOnOnOn 


^ 




















Tl-VOOO 

















S 







































t-^ H in ON --i-oo 

. moo 



m -^oo moo n 
vd o ■* ON moo 



I VO m O m 



m m -"l- -^ in mvo vo vo t-^ t^oo oo a^ On o O O 
ro"■)'ommm^''mmmmrommm-r^•^^■*•^-T(-■<J-T^Tj-T*■^T^-■<^•^^■ 



u 
a 
m 

x' 


w ino ino Ti-ON-^ON^ON^ON-cj-ONTj-o ino m-vo h t^ooo mov-^ovo 

t^O ^i-^M .^t-^H -t-QO w u-oo M in On mvo m t-^ -ir t^ h T^oo h in o- N 

U-) in in in m in mvo vovovovovovovovovovovovovovovovovovovovovovovo 
mcommmmmmmmfommrommmmmmmmmmmmmmmmm 


1 


mvo On n u-.oo n moo i- ■* t^ ■>*• t^ mvo O mvo On w vo On N moo 0) moo 

(N m tJ-vo t-^oo d " N -"i- mvo oo o 6 in m ^vd r-^oo on h ci m invd t-^ on 6 i^ 
000000-'-'^""-''-'-'iNN!NNN<N(N(NmmmmmmmT)-T(- 

(N(NCNll.Nl<N)tNiMW(N<N(NCS^(NCN,c^(S(NNCONN{N(MNNNNNf)<S 
OON(N(N«W(NINN0)P4NCNtNO(N(N<NCMWNNNNN(SNN0IN 



Q 
O 

a 
u 


VO t-^ 1^00 ooovo-o *■• H H f) IN mmm-<i--<»-mm mvo vo vo ^^ t^ t-^oo oo oo 

8!s!8.naaan8!8!anHlH8^aa8;al8NHf a 




N -B-VO 00 N ■'l-vo 00 (N) -^VO 00 n N .*VO 00 N ■<^vo oo O N -^vO 00 o 

„ „ M M H o< (N n .N CI rocommm-^-^^-i-^mmmm mvo 



FUNCTIONS OF A ONE-DEJRSE CURVE. 



323 



2 

i 


N 


■<^\O00 N -*-^ 00 N -^VO 00 N Tl-\0 oo N -^vO OO N -^^ 00 


< 


t~~ ■* 


woo loroo WiON l^rnN t^inrOMOO^ ^N OoovD T^(N Oi^ 


roo) 


rOt-^N t^MVO mVO m mo LOO ■*0\-*-0\ roo3 roco r'. t-^ n t^ P) r^ - \0 

r^Lf-,t^t^t^t^i^r^ t^oo 00 do CO 00 CO CO oo'oo^c»'oo cJ^oo'co oo oo~co oo oo 




H O 


t^ IT) ro M o t^« Tho Ovr-m^N M ooo « m -J- ro M o o-co t^vo m 




rOVO "^00 N inOrot-~M tnoo P) VO 'l-OO « mO^rot^M mOv<NVO •'l-OO 

1 ^ To s^ ?; 1 H H ^ i H ^ 1^ n ^ ^ ^ I'm ?i II 5- ^ ^ 


Q 
O 
Q 


t- 


T^t^0 'i-c^w T^^~M T)-53 M moo - moo Pi m o (n \0 o rovo O rn t^ 


5?? 


■* m t^oo On M M -n mvo r^a^Ow^o^mI ooo-Mpjro mvo co o p) 
(N M n c< (N rn ro ro ro n n ro T^ Tt- ^ rf rj- Tf -^ Tl- m m o L.T m m invo O 

C? C? S' C? f? C? C? n'^ N^ N^ N C? P? N^ (N^ P? C? N^ C)^ P? Pi ' P? P? P? P? P? pT n' P? 



p4P)P)Ht-(Mi-iMM000O-ONOv O^OO 00 00 t^ J^ l~^\0 "OO iom-<J-Tj-Tj-fO 

P) -^^ 00 O P) -^^ 00 o PI ■+mf^ONw romh^CM nmt-^o^H romr>.ONi-i 
M M H M PI PI p) PI PI ro ro r -1 n ro m Tj- Tl- -)- rj- -*- ! -) I o I n m lovo >o \0 vo vo t^ 

P) PI W P) P) PI P) P) PI P) N P4 PI PI PI P) Pt P) Cl -1 CJ <•! PI -1 PI po p) p) cl PI PI 
O0^0^0^0vO0^O0^CJ^0N0^0N0^0^0^0^C7^OOOC^C^O0^0^C^0^O0^0^ 



N ^^ O30P1-4-O000P1 ■^'O 00 O P' -h'O 00 O P) T)-if^ 00 '^ PI Tt-VO CO O 
M M w -I H p) p) PI P4 M nmronro-^-^'^-^-^mmmm lo^ 

O N -^VO CO PI ■+'0 00 M ■^"O 00 O N ■+>0 00 O P) -^^ 00 O PI -^^O 00 O 

M M w M M p) pg PI PI PI mromnm'*--^Tj--^-<*-mmmm m>o 

O'O P»oo -+0 r-roompioo m^oo rt-i-i r-.'^-w t-^-<^Moo mwoo mpi o t^ 

Tj-oo rot^N r^H>0 O mO •* O ■'l-co noo M t>~pio w'O O mO -^Ov'^on 
O o w " PI M rom-*-*mm m^o vo t^ t^oo oooo\00-wpipipiroro->i- 



0\0 PO Ov>£) ro O P~ ■* w 0>0 POO P^mP) O t~.-hpl 
^1-00 PI m On ro t^ O -"l-oo m m o> fovo O -^oo 



O On On O 

P^ t-. P-vOO oo oo oo CO oo o 

coroc<^rOrOPnrOPOPO( 



n ■* -^ m 1 

00 00 00 0" 

1 m m po I 



O 00 NO 
■*oo M 



. to ro m ro I 



CO ro ro ro n 



VO On rONO On poo On fONO On ponO On rONO ON POVO On ponO O roNO O ro 
H PI ^ mvo 00 On o PI po -^nO p^oo O >- p» ■* mvo oo cn - pi ro mvo 



ONONCNONf*^ONON0 o b 

PI PJ N N PI PI M roropOrororororOPOrO' 

P)PIP)P)PIP1P)P1PIP1PIPIP1(^PIPIPI( 



6 


T^TJ■T^T)■^T)-T^T^T^-TJ-■*T^^Tl-^T^T^^-*■T^'^T)-ropo^o-Poro^o.pl pi pi 


o 


^^^%^^^^^ ^?i^^^^cgo^S§§ aSNS;^=§.8 S t'^c'^o 2 :; 


OnOnOnOnOnOnOnO-O^OnOnO-OnOnOnO- OnOnOnOnOnOnOnOnCJnOnOOnOnQnO 


2 

* 


P) T^NO 00 PI -1-VO 00 PI -^NfN ro 'N f, ^\o 00 M -^NO 00 0- P' -^VO 00 o 
,- „ w M M PI N PI N PI r1rorororo^T^T^T^^mmmm m'O 



324 



FUNCTIONS OF A ONZ-DEdr.ZE C'JR"E. 



2 
S 


N ^OOO (N Tj-O CO O M -^O OO N rfO 00 P) rhO 00 (N ^O 00 

H H M « M c-i CI CI c^ N .-0(^mronTi-^Tj-Tt-ri-mmmm mo 


d 


C< M H 0^ O O\00 000000000000000000000000 OOOO H H N (V) m 


roco mcx) mt^tN t^N t^-i i^n i^n t~^c^i t^cM t^ci t^N t^ roco r'^oo moo -o 
m m Ti- -^ m mo o t^ t-.c» (»c3nO0 ^ " (n rs romT^^m mo o t^ r-oo 


<;' 


oooooooooooooooooooooooooooooooooooooooooocooooooooooooooooooo 


u 

Id 


37-4 
41.4 
45-4 
49-5 
53-5 
57-5 
61.6 
65.6 
69 7 
73-8 
77.8 
81.9 
86.0 
90 
93.1 
98.2 
'02.3 
'06.4 

14.6 
18.7 
22.8 
26.9 
31.0 
351 
39-3 
43-4 
47-5 
51 7 

60.0 


M 


■*-J-r^^■^T^Tf■<^T^T^^T^^Tl-^,t-TJ-T^Tt-T^■^^^T^T^TJ-Tt-T^r^^V 



O O rnO O -^ I 



N N M N N 



00 t^O o m ■ 



00 P! m On N O O 

'j-o c^oo o H m 

M1-IHMCMNN(N 
Mf<NP4N(NN(N 



O O m t^ w •<!- 



M o On 000 t^o o m Tf m I 

On - N T)-0 



o 0^00 



o^-tmint^OMromi 

N rnmrnrnro-*-^'*-Tt--<i-mmmm mO O O o o f^ K 1 

m '"O rn rn en ro CO m en rn rn ro CO ri") m m m m ro rn ro CO CO c 

OnOCJnOONOnCJnCDnC^nCJnOnOnOOnOnONOOO^OnOnOOnOnOOnOnOnOnOnOn 



O N -^O 00 O N -^O 00 O 



cs N CNi N N m m ( 



■*•<l■■*•<i-■<^mmm1 



O PI Tho 00 



t^o in ■* •* m ( 



000000 



rvo o m m m ■ 



o o -^00 N inONrnt--H in "\ 
P4 m rn c. ^ 'J- -*• m mo no o 
000000000000 



I On rn r^ M m o^ ro tv 
O H M (^ PI CNi t^. ro 



■* mo 00 o O N I 
, r~» t^ f^ r^ r^oo 00 

I rn t^ ro rn c^ rn rn I 



O c^O O ro t^ ■>*• (^ M -^oo M m On N O 
1^ ON o H rn -^ m t^oo ON M pj 

000 ONO*OnC3NOnOnC>00 o 



O O On onco 00 t^ t^o o m m ■*• ■ 



m m t^ <3n « mi 



O On 0-00 ( 

N rl m t^ On 



D PI T*-o 00 Q P) -^O 00 O P) -^O 00 
. t^OO OOOOOOOOOOsONOnOnOOOOOi-I'-'"!-!-! 

p) Pi N PI P) P) P) N N p) pt N N P) PI mmmmmmmmmmmmmrofnro 
C7NOC^ONC^C3NONC^ONO^ONO^C^O^O^ONONONONO^ONONONONONO^O*C7NC>0'0 



o P) ■♦o 00 o 



FU 



CF A CNE-DEGREE CUhVE. 



325 



1 

1 

1 

H 
H 


Q 


N ^vooo N ^OOO N ^vocg 0^ ?^ ;?;%~ 0. N ^O 00. N j;0 c» ^ 


<N •♦o 00 p) lo t-^ OS H ^o o\ M ^\o o\ o ■<^ t-- rovo o « inoo M -r^ t-^ « 


t-. f ) f-~ <S CO coco <r>CO 'TOv^OMOO inO^ MVO (N f^N r^ rooo ro o\ -*• OviO 
fO T}- -»■ Lo liTO O r^ t^co 00 ONO\0 H w M N roro-<J--Tir> lovo O i-^ t^oo 00 O 
ro ' T ro r«) ro iO to ro ro ro OT n ro ri- -<j- T^ T)- T^ Tt ^ tC T^ T^ Tf ^ TT 'i- -<f ■>*■ ■* ■*- 
OOCOOOCOCOOOCOOOOOCOOOOOOOOOOOOOOOOOOOCOOOOOCOOOOOOOOOOOOOOOOO 


w 

CO 


■>j- t^ rOO 0\ N toOO M moo H moo M iOOsCNVO rOt-~M u^O^^Ot^M lOO> 

vo mosrot^pjo voo toco NO i-i m cTi ■♦oo ro f^ w vo' ■♦OS rooo N \0' 
ooososasOO«HN(NCJ(orOTf-^mm ^o vo ^. ^co.c» o; o; o O O ^ « 


d 

o 

Q 


mospivo ■♦r-^M mosNO q -♦co m m o ro^o q ■♦oo n "p q^ ^^ "^^ "_ "^oo. 

WNNP)NCqNPlcTpjNN(NNO.N«NNP)NP)NNr)NNP)MP<N 


Q 
K 
O 
31 
U 


m ■♦ ro M osoo f^O ■♦ ro N w 00 f^vo lO ro P) h 00 l^vo ■♦ ro N Osoo 


^^<=^ Ss S IS ^ SJvS v?^^^ ^ ?L ^^=1^^ eg oS-i'J^cg^ Ss S? S S;^ 8 

O^0^0^0^0^0^0^0>0^0^0\0^0^0^0^0^0^0-a^a^0\0^0^0^a^0^0^0\0\0^a^ 


2 


O p. ^VOCO PJ ^OOO N M-O.^ 0^ N ^♦O <» N. ;JO 00. N ;*^00^^ 


h 

H 


2 


P. ^^00 P. ^Oc» CO ^vOc« C « ^OC« O ^^OOO N J;Oc»^ 


Q 


(^ ■♦ ■♦ mvo t>.oo On « N ro ■♦ mvO 00 Os P) ro mso oo Os m ro lovo 00 O P) 


rooo rooo rooo rooo Tl-0\-^0\^Os'^asTfO mo mo moo mmd mo n t>. 


X 


O w f»1 ^O 00 OSM romc^OsM comt>.ONw ^O oo h roo oo m poo On N ■♦ 

ill iitif 1 1 II m 111 i i ii mil § III 


Q 
Q 


-♦00 M mosp»0 O rot^M >*-oo NO On ro r-^ o -♦oo m m 0\ fOO ■♦ t-^ m m 


ro ■♦o t-^oo >-i ro ■♦ m r-^oo On m N ro mo co Os a P) fo ■♦o tv o\ o m ro ■♦ 
^-^-^^^^^-^-^--^----^------icooo.^ 

NP<NNNNNC<NNNPINP)NPINNP)MM<NP)NP)P<NNNNM 


Q 

O 

X 


m ■♦ ro PI M o Onoo t^ t-o m Th ro PI h onoo two m ro N m o 0^00 r^o m 


c^cg^ 5, S S^ S;^ 8 S 3"^ •§ 2 2 ;?^ <S 2^ J3 J? J? &• 2^ ?^ ?;? ^S^'^ ^ ^ ^ 


9. 


P. ^OOO PJ ^♦OOO P; ^^~ ?^^^~ N ^0=0 N ;j;0«^ 

i 



326 



FUXCl'DNS OF A ONE-DE(J 



CUR VE. 



H VD O 

t^ N 00 



O "O M vo N t^ rnoo Tj- o\ 



f^ t^ r^ r^ r^ I 



N 00 -^ O \0 CI 00 

0000 O-OO- ■- N 
- f^ t^ r^ t^ t^oo 00 00 00 00 



VO N 00 ■* 0\ u~ 



roOLOMOO -^O^O 



1 P) 0\^ M OVO ro O t^ 



M VO O lO a^ "^ <>» '■^.C 



■^■*Tt-Tj-,r-,j--^xt- 



ro ro -^ ■*- in u-.vo vO 



T^oO N VO O T 



M in o ro t^ I 



-^ Lo t^oo 0^ ^ w -^ lOVO ( 
t^ r^ t-^ t^ t^oo ro 00 CO 00 ( 
inioinioininmiomu, 



O ON t^ in 



000 r^inmH ooo\o -rro 
M ro m t^ O M ro ■^^^0 00 O 



00 O^ 0\ 

ii")ininLntnLninLnmininmininu"-minininintnLnin mvo vo vo ^ ^ O vo 
0^O0■*0^OC^O0>0^0^0^0N0^0^OO0^O0^0^0^0^0^O0^0^0^0^O0v0N 



O N -^^ 00 O ts -^vo 00 O f) -^^ 00 O (N ■'f^ 00 O N -^VO 00 O M M-VO 00 
,- H H « M c) w N P) C4 mror^^mm■^■*■^T^■*u^^nlr, in mvo 

O t) •<*->0 00 N -^^ 00 O M -+^00 0(N -*vO 00 O M "S-'O 00 O f tO 00 

M HI M 1-1 « N c^ pj c4 N mmmrnm-*-4-'<r-<(-'rinLnuju-, m^o 

« '^fOO ^ -^OO M in On C< O O -^OO M VO O -"l-OO N NO O in On moo N t^ W NO 1-. 

.Ad in - NO - t~- M t^ r<-oo -^On^O inMNO m r-<N00 n-,00 4- On in O no m t-« 
OnC Oi-iwc^Nror'-. Tj-Tj-ini/Nor^ r^oo ooo^ONOOMHiN<Nro-<f-*inin 
■^t- ^ iri in in in in in in in LTi in 1' in in LO in w in inNO noncnon^nonCnononono 
oooooooooooocooooooooooooooooocooooooooooooooooooooooooooooooo 



On -^00 N NO >-• in O -^ On f^oo fooo 
in o -*■ o> rnoq 

in in 



ooni>.'-noouion -^oo rooo N 1^ - 
oooONOOi-'-i-iWMr'-rorhTi-in 

in in m.NONONONONONONONONOVONONO 



00 N no O "^ tv 



I On ro t^ "*-00 N NO O ■* t^ I 



in m ir> m m 1 



On M N ro inNO fX3 On O M 

m'*'>1--<f-<l-'^'*'*iOinmin 

loinioinininini/iiniviinin 



On r<-. tv - inoo 0) NO 
-id r^-ONO >- rO-<i-Ni 



Q 


00 NO 


m -<f (s 




On 00 


.in 




(N 


M OnOOnO 


in roN 


On tvNO 


^ 


m M M NO m rf 


C 


8S 


f^^ 




u 








IH 








m 




















































U 


ONONONONO'ONONONONONONONO^ONO^ONONONONONONONONONONO^ONONONONON 


z. 


N 




c 




ThNOOO 


o 










S 

























FUNCTIONS OF A ONE-DEGREE CURVE. 



:Q7 



O N ■^^ 00 O D •<«-^ < 



troro■*T^■^t•■*■^^lo^i^lDu^ »ov< 



O 00 t>. ID ■* ( 



00 t^\0 »riij-f«-,c<wi-0o^ 

O ^ woo •^^0^ 0)00 ■*0^ 

ONOoooooodooooooooooi-"M^-(H>-fH^MMM 

00 O^C>O^O^ONO\0>0^ a^O^O^O^(^O^O^O^O^C^O^OO^OOO^■0^000^0^0 



o^vo -<^ N I 

in w t^ ro o^ 



o "o 



o osoo t^vo 



O O O^O^C^O^O■C^OO^O^O O 



■* O ■* o -^00 rooo rooo trioo rooo moo moo moo N t^w t^N r^w t^Noo m 
m m ■>!■•<»■ lo invD ^o t^ t-^oo ooo^o^oo>-|'-'P^PlmmT^■^m lovo >o r^ t^oo 
o^o^o^o^o^a^o\oo^o^oa^c^a^ooooooooooooooooo 



■rf-^'^'<f'^*^'^'^"<^'^'^-n-'^ioiOLOi 



lO lO IT) lO lO LO IT) I 



Q 

o 

d 


mtvM >no^mf-,« uioN -^00 n vo o ■<*-co m\o o loomt^" uiomt-^w\o 


« « 'J- invo 00 (^ " <N m lo^o oo o " N m lo^ oo o n m i/^\o i^ o o w m 
ID lo lo lo lo 'o u^vo >o\ovovo'0>o t^t^t^t^r^f^ t^oo cooooooooooo oo>c>. 


Q 


mwoo« T^N OoovO ■<*■« ot^mmooovo -<j-w o^^«u^(^^ ooovo mw o^vO 


O 


?^t tl^^l;l^i^ttH>ll!f l^t tf S^S^S;S^S;S;S;i:S^S;S; 



O N -iJ-\0 00 O N -^VOOO O N -^^ 00 O N -^VO 00 O N -^-^ 00 O M -^^O 00 O 

M r- 1- M w N N N N cj mmmmm->»-->i-->*--«t--*ioio'-oio lovo 

N -^^ 00 O N -^^O 00 O N -^VO 00 O « -^VO 00 O W -^^O 00 O Ci 'i-^ 00 O 

M iH 11 iH iH PI CI c) o M mmmmm-^-^TtTj-rj-ioirimio \r,\o 

■*i-i t^movo N ouiM o\iop) ovo mo c^-*mooo mooo irimooo irjm 

mcJ^■<^o^ 1-1 t-~-Noo -^O'-niHvo noo •>!^o^>nlH\o woo Tfosm"- t^cMoo ■* 
w w m Tf -^ cr^ lovo o t^ t^oo oo\0 O ^ t-* N mm-'i-trio lovo t^ c^oo oo o 

OOOOOOOOOOOOOOOOOOOOOOOOOOOO OSO\OnOOOnOvOOOO-OnOOOOiO\ 
OOOOOOOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 



t^ IT) M 0\V0 •* M OVO -*- ONt^-*N 000>O Tj-N O OtvinmW O t> I^VO ■* 

- in o •* o^ ■* <> moo' moo' n r^ <n i^ n \o m o i- •■O O lo O in C in o> rj- oi •<»■ 
~ * " "■ — ■ . - O ■-! -i M N N m 

' 0\ Ov O O^ 0\ On O^ 



ivo O -+00 e»\o o -<too Nvo -^oo N\o O -^-oo N^o -"j-Ovmt-^M inoNi 



m Tt-vo t^ On I 



m -^vo t>.oo o 



Tj- in f^oo 0\ ' 



N W (M N W (M 



tN.inmi-1 ost^mmi 



»- m, -^"Ooo N minr^ONw « •+no oo O *' mint^oo n -^no oo o^ h m lo 
M -I n I- 1-1 p) N (N N N p) mm(v->nm■<^■<^■Tt■■^}-1«--^lnln^nlntn mvo >o >o 

\OVONO\ONO»OVOVOVO'0\OVO\OVONO\OVOVOVONOVO\O^VONO\OAO\OVO\0\0 
0^0^0^0-O0^0^O■C^0^0^0^0^C^0^0^0-0^0^0^0^0^O0>0>0^0^C^0^0^0^ 



O N ■*NO 00 O « -^NO < 



O N 'f\0 00 



rommmmTr-*-^-<i--^ioi 



328 



FUNCTIONS OF A ONE-DEGREE CURVE. 









o' 

H 




M -^VOOO <N Tl-OOO CM -<J-vO 00 (N -4-^0 00 O N -^VO 00 (N -^no 00 O 

„ „ „ „ „ c< CM (N IN CM rororomro-«--«--<j-^-<i-ioioir)in mvo 




Q 
< 

u 

w 

C/3 
X 

Q 

o 

a 


lovo t-^ ON M ro -^ in tv. o n tJ-o oo c< m-nO oo O o-, m i-^ O cm inoo O ro 

6 NO csoo' lA"' t^roc3Nir)H00 Tt-ONO c^' OniAh t-^roOvo CMod lOH t^mOvo 
in inNO NO t^oo 00 OnonO i-i I- CM roro-«--^ lono no r~.oo ooooNQi-'i-iCMr'ro 
ro f, m <o ro t 1 n-) r<) n Tf- ^ -r TT Ti- Tf ^ rr -J- rr Tf rr rt- TT rr r^ m in in u. ui u-i 
OnOnOOnOnC?nOnOnO\OnOnC3nO>OC3nC3nOnOnC>C7n<3nC3nOnOOnOnC?nOC3-OnC> 




NO 00 M "^'O O M i^NO On M in t-^ rovo On CM moo N idonm inONrot-^O •* 

NO M r^ CN r^ CM t^ rnoo rooo Tf o Th d mo' in -' no « t^ cn r^ r^ocj ro On ^ 6 m 
ro Tf ^ 1/ 1/ , NO NO 1-- I--0O oo ON On C H « N rj m f, rr ^ m, m.vo NO tv. t-~00 ON O 

inmmmmmmmmmmu-iminmmmmuTi'-mu-, mmw. mi/, mSSS 




M m o ^00 CMNO mONror^CJNO ThONCOt^Hvo -O-oo ro t^ m no O -^oo 




NO t-^oo 1-' ro .^NO t--oo w ro -^no t^oo w ro ■<^NO t-~-cx5 o >- ro t^no r^co 
rororo-<r-*'a--^-*-«r-*mmmmmm mNO nonqnononono t-^t-t-^c^t-^t^t-^ 

CM(NNP)N(NCM(NC<)CMCMCM(NC^CMCM(NCMCM1N(NCMCMNCMCMP)M!NMN 




Q 

o 


n-NQ t^mM ONt--^N ONNO ThMOONO roo (^rno ONt-'i-woo mroo f^-<(-M 




t-~t^t^r^t-~t--t-~i^t--t--t-^r^t-^i^t^i-^ t^oo oooooocoooooooooooooooooco 
0^0^0-C^NONC3NONC3NO^O>C3NONONO^aNO^ONaNOO^ONOO^O^O^C3NO^O^C3NC3NLA 






r. ^^00 CM ^NOOO CJ ^NOCg N ^NO C» ^ :JVO 00 i;, MJVO 00^ 




1-1 


z 

i 


O « ^Nooo CJ ;^^oo cj ^NO<« 0^?^pj;^%0.« ^NOO^O « ;^NO<»^ 




< 


On Onoo 00 t-^ t-.vO NO vo NO NO in mNO vo NO NO NO r-- t--.oo ooONtjNOHHNro-^m 

ONmHod rocJ\mH t^fOo>mH t^roONini-I i^rocJNin" t^4.o'No' cmoc> 46 
NO t-.oo OO OnOnO « -H (N CM (0 1--*- m mNO t>. r-oo co on o ►' cm ri ro --^ -r > i 

ONONONONONONONCJNONONONONONONObNaNONONONOONONONSoNSSSo^O-N 






X 


OOwHNNro-*-<t- mNO t^oo o M N ■>*- m t-^co h ro -^vo oo N •'I-no 

rooo rooo rooo' rood rooo roco roco ■<^ON■>J-o■>*■ON-1-d in 6 >od io>-vd i-vo 
00 S ONONOOMHCMCMrorOT^T)-mmNONOt^^-ooONO^OO"MPl(^ror^ 




Cnmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm 




O 

Q 


NO ■<^00 CMNO H mONrot-^HNO O -*-oo <nno m mONrof-NNO -*oo ro t^ h 




IIIIIIL^IL^L-L-.-L^:^L^L^IL^I^i^l 




Q 

O 
X 

u 


NO ^<S cjNt^iJ-N t^-mroOoo mrooco mroOoONO roOoo mroooo mro 




00 o CM romtvONM CM -*-no oo o - ro mNO ooOMromr-^ONON -^no r^ on w 
H CM M <S CM (M CM rorororororOTf-Tj-TfTj-Tj-mmmmm ir,'^ vo vO no no NO t- 

ONONC3NON(?NONONO\ONONaNONONONONO\C3NONC3NONC7NCjNONOONCy.ONC7NC3NONON 




z 


M ^No CO pj ^Nooo N ^vo <» 0^ N ^Now o N ^ivooo p^ N ;^^c»^ 







rUyCTICXS OF A OXE-DEGREE CUEVE. 



3:9 



O O TT^OO O N "^MD CO O N -^VD 00 O N ■^'O CO O N -**o OO O N -t^o oa O 
« -, .- w _ c) f) o N f) ^or'^roI■oro-*■^Tl-Tl-•^^u^u^w-)u-) in\o 

vOwvo mvOw^ mvOhvo CJ f^ fooo t)-0\0 h t^mOvir^MOO rj-Q P~f<10 t^ 
f^-t-O t~xnCMD r-~i c^\0 CI OiOC^CO lOMCO lOMCO -1-HcO -+1-100 ^"OO f 

p) f. 1- ■+ i.jvtj ^u t^ t^cx) o o- O '-' - CI fo n -"i- K") >ovo f^ i-^oo 000 - w ci 
t^r^t^r^f^r^t-^r^i^r^t-. 1^00 or)oocxDooooooooc/-oooocooooooo oooc^ 

O^O^O^O^O^O^CJ^O*O^O^O^Q^CJ>C^C^O^O^O^O^O^C'■O^G>O^C3^0'OC^OlO^O 

00 -"I-OVO 000 -1-0^ rOOlANOO lANOO u-)0 0^'<:i rOMOO inp) t-^iop) O 

C\iOM^C '1 t^rO'T\-rO 10 " r^NOO -:^o^lnl-'0 0)00 -l-OiO" i-^MOO '^0 
lo^o t^ ^^oo oDoo.i-i-tsciro i^-t lova ^ r^ t^oo coo^OO^-|-lCNf--l 
»o 10 in "O in 10 Lo tovo \cvovo^ovovovovovo^cvovovovo*ovo t^t^t^r^r^r-^ 



5 


00 




O-w^O 


Tt- or^ 


I-. N VC 


H in 


0^00 


CO t^ 


H vo 


. 


inocn 


00 PJ t^ 


M vo 





M 


r^ 






00 






PI 






PI CO 1 1 
















T^ ^ -+ 


^ 'I 










2 


00 


Y1 


























cs P) P) 


(N eg cs 


(N P) 












PJ P) N 


p) PI PI 


Q* 


CO 





1-^-001^ 


rj- H CO 


10 N CO 


.nP, 


c^ >i PI 


OVO 


r-l 00 


CO vo 


C-. Cvo 


CO (^ 


S 


,1- 


^ 


r^ - t-1 


-r ~ r~. 


C^ - P4 


5^ 


h^rr, - 


S "5 - 


8^ 




M ro ^ 


\0 00 ON 


M ro -J- 


00 


-L 


COOOCOOOCOOOCOOOOOOO 






f^^ 


000 


000000 





0\ C7^ 0^ 


000 


OOC^OC3^C^OOO^O^C3^C^ 




OC3^ t> 0^ cr 


000 



n PI , -f-vn 00 



O Pi -fO 00 o 



pj -^vo 00 o 



fO'O o PI 1000 H inco 



CO -r ^ invo 'O r^oo 
0000000- 



10 o rovo ^00 

CO -r M I-^ -:_•- o VD 

" PI CO CO -+ ir> I -)' 

o 10^ vo vC ^ ^O >0 vc 'O vr> ' 



O O O M 
. 1 in 10 10^ vo _ _ _ 

oc- 000000000 



-5h03 N O O -+00 

10 ^ O " t-^ PI t^ 
O O O H i-' PI M 



CO t-^ " \D mo 

r-.OO -r c- 10 o \o 

n CO -1- Tt- ir-AO ^ 

in 10 10 Li-> iTj ui in 



"^O-l-OTt-O-^O^OinO^ 
00 n r^ 



>o o -1-00 cot-^wvo -*ocoi^pivo o mo cooo N \o m in o ->*-i 

-] -i-O l~-00 O >-' CO -hvo r^<X) O - ro t!-vo r--oo O m co -j-vo t-^00 O 

I 0-. OOCOOOOOC^OOOOOOOO 00 own >•--">-' i-^P) 

-t^i-^t^t^t-^t^t^t-^t-^i-^ f^o- cocooooooocncoooooooooooooooi 

PIPIPIPIP)(NPIP)PIPIPIPIP)P)P)P1PICIP1PIPIPIPIP)PIP) 



M OVO CO ' 
-' -ho CO ( 



rOOVO CO O t-^ 

-vo 00 O P) 






in PI 0\0 CO t~« -* 1 
O PI CO 10 t~^ o O 



1 vo vo vo \o r^ 



OVO CO 
6 pi -i- 



TABLE XVII. 
RISE PER MILE OF VARIOUS GRADES. 



332 TABLE XVII.-RISE PER MILE OF VARIOUS CEADEr 



Rise 


Feet per 


Rise 


Feet per ^ 
Mile. c 


ise 


Feet per 


Rise 


Feet p r 


Cent. 


Mile. 


Cent. 


er 
ent. 


Mile. 


Cent. 


Miie. 


.01 


.528 


.61 


32.208 1 


21 


63.888 


1.81 


95-568 


.02 


1.056 


.62 


32.736 I 


22 


64.416 


1.82 


96.096 


.03 


1.584 


.63 


33.264 1 


23 


64.944 


1.83 


96.624 


.04 


2. 112 


.64 


33.792 I 


24 


65.472 


1.84 


97.152 


•05 


2.640 


• 65 


34.320 I 


25 


66.000 


1.85 


97.680 


.06 


3.168 


.66 


34.848 I 


26 


66.528 


1.86 


98.208 


.07 


3.696 


.67 


35.376 I 


27 


67.056 


1.87 


98.736 


.08 


4.224 


.68 


35.904 I 


28 


67.584 


1.88 


99.264 


.09 


4.752 


.69 


36.432 I 


29 


68.112 


1.89 


99.792 


.10 


5.280 


.70 


36.960 I 


30 


68.640 


1,90 


100.320 


.11 


5.808 


•71 


37.488 I 


31 


69.168 


1.91 


100.848 


.12 


6.336 


.72 


38.016 1 


32 


69.696 


1.92 


101.376 


•13 


6.864 


•73 


38.544 I 


33 


70.224 


1-93 


101.904 


.14 


7-392 


•74 


39.072 I 


34 


70.752 


1.94 


102.432 


•15 


7.920 


•75 


39.600 I 


35 


71.280 


1^95 


102.960 


.16 


8.448 


.76 


40.128 I 


36 


71.808 


1.96 


103.488 


-.11 


8.976 


•77 


40.656 I 


37 


72.336 


1.97 


104.016 


9.504 


•78 


41.184 I 


38 


72.864 


1.98 


104.544 


.19 


10.032 


'P 


41.712 I 


39 


73.392 


1.99 


105.072 


.20 


10.560 


.80 


42.240 I 


40 


73.920 


2.00 


105.600 


.21 


11.088 


.81 


42.768 I 


41 


74.448 


2.10 


110.880 


.22 


II. 616 


.82 


43.296 1 


42 


74.976 


2.20 


116. 160 


.23 


12.144 


•83 


43.824 I 


43 


75.504 


2.30 


121, 440 


.24 


12.672 


.84 


44.352 , I 


44 


76.032 


2.40 


126.720 


•25 


13.200 


•85 


44.880 I 


45 


76.560 


2.50 


132.000 


.26 


13.728 


.86 


45.408 I 


46 


77.088 


2.60 


137.280 


.27 


14.256 


.87 


45.936 1 


47 


77.616 


2.70 


142.560 


.28 


14.784 


.88 


46.464 I 


48 


78.144 


2.80 


147.840 


.29 


15.312 


.89 


46.992 I 


49 


78.672 


2.90 


153.120 


.30 


15.840 


.90 


47.520 I 


50 


79.200 


3.00 


158.400 


•31 


16.368 


.91 


48.048 1 


51 


79.728 


3.10 


163.680 


.32 


16.896 


.92 


48.576 I 


52 


80.256 


3.20 


168.960 


•33 


17.424 


•93 


49.104 I 


53 


80.784 


3.3ci 


174.240 


•34 


17.952 


.94 


49.632 I 


54 


81.312 


3.40 


179.520 


•35 


18.480 


•95 


50.160 1 


55 


81.840 


3.50 


184.800 


•36 


19.008 


.96 


50.688 I 


56 


82.368 


3.60 


190.080 


•37 


19.536 


•97 


51.216 I 


57 


82.896 


3.70 


195.360 


.38 


20.064 


.98 


51.744 1 


58 


83.424 


3.80 


200.640 


•39 


20.592 


•99 


52.272 I 


59 


83.952 


3.90 


205.920 


.40 


21.120 


1. 00 


52.800 I 


60 


84.480 


4.00 


211.200 


.41 


21.648 


1. 01 


53.328 I 


61 


85.008 


4.10 


216.480 


.42 


22.176 


1.02 


53.856 I 


62 


85.536 


4.20 


221.760 


•43 


22.704 


1.03 


54.384 1 


63 


86.064 


4-30 


227.040 


•44 


23.232 


1.04 


54.912 I 


64 


86.592 


4.40 


232.320 


•45 


23.760 


1.05 


55.440 I 


65 


87.120 


4.50 


237.600 


.46 


24.288 


1.06 


55.968 I 


66 


87.648 


4.60 


242.880 


■''I 


24.816 


1.07 


56.496 I 


67 


88.176 


4.70 


248.160 


.48 


25.344 


1.08 


57.024 I 


68 


88.704 


4.80 


253.440 


•49 


25.872 


1.09 


57.552 I 


69 


89.232 


4.90 


258.720 


•50 


26.400 


1. 10 


58.080 I 


70 


89.760 


5.00 


264.000 


•51 


26.928 


I. II 


58.608 I 


71 


90.288 


5.10 


269.280 


•52 


27.456 


1. 12 


59.136 I 


72 


90.816 


5- 20 


274.560 


•53 


27.984 


1.13 


59.664 I 


73 


91.344 


5.30 


279.840 


•54 


38.512 


1. 14 


60.192 I 


74 


91.872 


5.40 


285.120 1 


•55 


29.040 


1.15 


60.720 I. 


75 


92.400 


5.5c 


290.400 


.56 


29.568 


1. 16 


61.248 I. 


76 


92.928 


5.60 


295.680 


•57 


30.096 


1. 17 


61.776 I. 


77 


93.456 


5.70 


300.960 


.58 


30.624 


1. 18 


62.304 I. 


78 


93.984 


5.80 


306.240 


•59 


31.152 


1. 19 


62.832 1. 


79 


94.512 


5.90 


311.520 ) 


.6c 


31.680 


1.20 


63.360 I. 


80 


95.040 


6.00 


316.800 



ADDENDA. 



ADDEISTDA 



TABLE 

OF 
FEET, INCHES AND RECIPROCALS OF VARIOUS TRACK GAUGES. 



6' 


72" 


.013389 


Metre 


39.375" 


.0254 


5' 


60" 


.01667 


3' 6" 


42" 


.0238 


4' 9" 


57" 


.01754 


3' 4" 


40" 


.0250 


4' 81" 


56i" 


.01769 


3'0 


36" 


.0278 



TABLE 



MINUTES OF A DEGREE EXPRESSED IN DECIMALS. 



1 


.0167 


16 


.2667 


31 


.5167 


46 


.7667 


2 


.0333 


17 


.2833 


32 


.5333 


47 


.7833 


3 


.0500 


18 


.3000 


33 


.5500 


48 


.8000 


4 


.0667 


19 


.3167 


34 


.5667 


49 


.8167 


5 


.0833 


20 


3333 


35 


.5833 


50 


8333 


6 


.1000 


21 


.3500 


36 


.6000 


•51 


.8500 


7 


.1167 


! 22 


.3667 


37 


.6167 


52 


.8667 


8 


.1333 


1 23 


.3883 


38 


.6333 


53 


.8833 


9 


.1500 


j 24 


.4000 


39 


.6500 


54 


.9000 


10 


.1667 


I 25 


.4167 


40 


.6667 


55 


.9167 


11 


.1833 


26 


.4333 


41 


.6833 


56 


.9333 


12 


.2000 


27 


.4500 


42 


.7000 


57 


.9540 


13 


.2167 


1 28 


.4667 


43 


.7167 


58 


.9667 


14 


.2333 


29 


.4833 


44 


.7333 


59 


.9833 


15 


.2500 


30 


.5000 


45 


.7500 


60 


1.0000 



386 



ADDENDA. 



CONDENSED TABLE OF RADII 

INCLUDING SHORT CHORDS. 



Degree 








OP 


100' Chord, 


50' Chord. 


25' Chord. 


Curve. 








r 


5729.66 


5729.60 




2° 


2864.93 


2864.82 




3^ 


1910.08 


1909.91 




4° 


•1432.69 


1432.47 




5° ■ 


1146.28 


1146.01 




6° 


955.37 


955.04 




7° 


819.02 


818.64 




8° 


716.78 


716.34 




9° 


637.28 


636.78 




10° 


573.69 


573.14 




ir 


521.67 


521.07 




12° 


478.34 


477.68 




13° 


441.68 


440.97 




14° 


410.28 


409.51 




15° 


383.06 


382.25 




16° 


359.26 


■ 


358.17 


17° 


338.27 




337.11 


18° 


319.62 




818.46 


19° 


302.94 




30164 


20° 


287.94 




286.57 


21° 


274.37 




272.93 


22° 


262.04 




260.54 


23° 


250.79 




249.22 


24° 


240.49 




238.84 


25° 


235.65 




229.30 


26° 


222.27 




220.49 


27° 


214.18 




212.30 


28° 


206.68 




204.76 


29° 


199.70 




197.70 


30° 


193.60 




190.79 



ADDENDA. 



337 



To express gradients per cent, (page 332), in angular meas- 
ure, multiply the rate per cent, by 34.3; the product will be 
given in minutes of a degree. 



EXAMPLE. 



Gradient 

PER CENT. 


Minutes. 


GUADIENT 
PER CENT. 


Minutes. 


.20 


6.86 


2.50 


85.75 


.40 


13.72 


3.00 


102.90 


.60 


20.58 


3.50 


120.005 


.80 


27.44 


4. 


137.20 


1. 


4.30 


4.50 


154.35 


1.20 


41.16 


5. 


171.5 


1.40 


48.02 


6. 


205.8 


1.60 


54.88 






1.80 


61.74 






2. 


68.60 







If the gradient per cent, be multiplied by 57. 14, the result 
will be expressed in hundredths of a degree. 

SOUND. 

At freezing temperature, 32 degrees Fahrenheit, in calm air, 
the velocity of sound may be assumed 1100 feet per second. 
For lower temperatures subtract, and for higher add, a half 
foot per degree. 

The intensity of sound varies inversely as the square of the 
distance. The velocity varies directly as the temperature. 
It is nearly four times as great in water as in air; and in wood 
ten to sixteen times as great. 



AMERICAN AND FRENCH EQUIVALENTS. 

LINEAR MEASURE. 

1 inch = 2.54 centimetres ; 1 centimetre = .394 inches. 
1 foot = .3048 metres: 1 metre = 3.2809 feet. 
1 yard = 3 feet = .9144 metres; 1 metre — 1.0936 yards 
1 rod = 16.5 feet = 5.029 metres; 1 metre = 0.2 rods. 



338 ADDENDA. 

1 surveyor's chain = 66 feet = 4 rods = 20.117 metres; 

1 metre = .05 chains. 
1 kilometre = .6214 miles = 3281 feet. 
1 statute mile = 5280 feet = 80 rods = 1.6093 kilometres. 



AMERICAN AND FRENCH EQUIVALENTS. 

SQUARE MEASURE. 

1 square inch = 6.4515 square centimetres. 

1 square centimetre = 0.1550 square inches, 

1 square foot = 0.929 square metres. 

1 square metre = 1.19659 square yards. 

1 square acre = 43560 square feet = 4840 square yards. 

1 square hectare = 2.4711 acres = 11960 square yards. 

1 acre = 0.4047 hectares. 

1 square kilometre = .3861 square miles. 

1 square mile = 2.5899 square kilometres. 

1 square rod = 272.25 square feet = .00259 hectares. 

AMERICAN AND FRENCH EQUIVALENTS. 

CUBIC MEASURE. 

1 cubic inch = 16.383 cubic centimetres. 
1 cubic centimetre = .0610 cubic inches. 
1 cubic foot = 28.316 cubic decimetres. 
1 cubic decimetre = ,0353 cubic feet. 
1 cubic yard — .7645 cubic metres. 
1 cubic metre = 1.308 cubic yards. 



^ 



INDEX. 



PAGE 

Abbreviations explained ix 

Acres, roods, and perches in square feet, Table VI 152 

Adjustment and use of instruments 23 

Angles of frogs, to find 129 

index, to find , 69 

intersection, to find 55 

plane 12 

to read on verniers 43 

tangential and deflection 50 

of switch-rails 130 

Apex distance of curves, to find 52 

Arc, functions of, to find 13 

Arithmetical complement 6 

Axemen, duties of 84 

Azimuths of North Star, Table 11 150 

Barometer, levelling by 29 

Bench-marks, proper intervals for • 83 

Bubble, to adjust on level . 25 

to adjust or transit 40 

Chain, to lay out curves with 63 

Chainman, duties of 42 

Chief engineer, duties of 79 

Chords, to calculate 54, 58 

Table XVI 269 

Circle, propositions concerning 49 

Circular arcs to radius of 1, Table VT 152 

Complement of an angle 12 

arithmetical 6 

Compoimd curves. See Curves. 

Contour maps, utility of 85 

Correction for curvature and refraction in levelling 28 

Cosines defined 12 

Crossings, plain rules for laying off 139 

Cross-hairs, to adjust 24, 26, 40 

eccentricity of 24 

to put in new 44 



340 INDEX. 

PAGE 

Cross-sectioning. See Slope stakes. 

Cubes and cube roots of numbers, Table XI 161 

Curves, circular, on railroad defined 51 

to find radius, length, degree, apex distance, chord, mid- 

ordinate, and external secant 53, 56 

form for field notes 70 

Curves, how to lay out on the ground, — 

with the chain only 63 

with tiansit and chain 66 

hints as to field-work 82 

protractor for 84 

slackening grade on 87 

terminal 88 

Cui-ves, simple, location of, — 

how to proceed when the P. C. is inaccessible 93 

to shift the P. C. in order to strike a fixed tangent 96 

to change radius from same P. C. in order to strike a fixed tan- 
gent 97 

to triangulate on 94 

to pass through a fixed point . 127, 128 

Curve«, compound, — 

how to proceed when the P. C. C. is inaccessible 95 

to compound a curve in order to strike a fixed tangent .... 98 

to shift a P. C. C. in order to strike a fixed tangent 99 

summary of rules for 101 

to compound into a tangent intersecting main curve on concave 

side 102 

to compound into a tangent intersecting main curve on convex 

siae 103 

Curves, reversed, — 

parallel tangents, radii equal 115 

parallel tangents, radii unequal 117 

angles unequal, radii equal 119 

angles unequal, tangent points fixed, radii equal . . . . . . 120 

divergent tangents, radii equal, advancing towards intersection . 123 

receding from intersection 124 

to shift a P. R. C. in order to strike a fixed tangent 125 

Curves, miscellaneous, — 

elevation of outer rail 141, 142 

degree of, to find by calculation 52, 55 

to find on ground 145, 146 

to connect curves of contrary flexure by short tangents ... 89 

to locate a Y from a tangent 103 

from a convex curve 104 

from a concave curve 106 

to locate a tangent to a curve from a fixed point 108 

to two curves already located 109 

to substitute a curve for a tangent connecting two curves ... 109 

terminal curves 88 



INDEX, 341 

PAGE 

Curves, miscellaneous — continued. 

trackmen's table of curves and spring of rails 143 

vertical curves, to calculate 36 

to project 39 

Datum in levelling 27 

Decimals of an acre per 100 feet for various widths, Table IV. ... 151 

Deflection angles and distances explained 50 

to find 57, 64, 68 

short rule for sub-deflections 68 

limit in field-practice 82 

Degree of curve, to calculate 52, 55 

to find on ground 145, 146 

Deviations from project admissible on location 81 

Distances, tangential and deflection, defined 50 

table of 155 

of frogs from toe of switch 130, 132 

tables of . 135, 136 

Elevation of outer rail on curves 141 

table of 142 

Excavation and embankment, to stake out 30 

External secants, to fiud 54 

of a r curve. Table XVI 269 

Extreme elongations of North Star, Table 1 148 

Feet in decimals of a mile. Table VII 153 

Field-work, suggestions concerning 79, 85 

Field-book, form of, for level 27 

for transit 70 

for slope stakes 33, 34, 35 

Frogs and switches 129 

rules for angles and distances 130 

table of, switch-rails straight 135 

switch-rails curved 136 

plain rules for locating, switch-rails straight 132 

switch-rails curved 133 

on narrow gauges 134 

patterns for 134 

Functions, trigonometrical, defined 12 

logarithmic, of arcs, to find 14 

General propositions in trigonometry 15 

as to circles 49 

Grade, to slacken on curves 87 

rise per mile. Table XVII 332 

Grade lines, how to project on map 86 

how to trs.ce in field 81 

Heights, to find by barometer and thermometer 29 



342 INDE3L 

PAGE 

Inches in decimals of a foot, Table VIII 153 

Index angles, to determine 69 

Instruments, adjustment and use of 23 

Intersection angles of tangents, to find 55 

desirable to fix on ground 66 

Level, to adjust 24 

Leveller, duties of 83 

Levelling, art of 26 

by barometer and thermometer 29 

correction for curvature and refraction 28 

form for field-book 27 

rules for exact work 27 

rules for survey and location 28 

suggestions concerning 83 

Location, problems in field 94 

admissible errors on ground 81 

form of record for 81 

projects, hints concerning . . . , , . . . 84 

of terminal curves 88 

of a Y 103, 104, 106 

Logarithms explained 3 

multiplication by 5 

division by 6 

of numbers, to find 4 

Table XII 179 

roots and powers by 7 

Logarithmic sines, tangents, &c., to find 13 

table of, XIII 197 

Maps, contour, utility of 85 

notes for 82, 83 

not sufficient for intelligent projects 79 

Meridian, to establish 44 

by equal shadows 45 

by North Star 45 

times of passage of North Star, Table I l^^ 

Multiplication by logarithms 5 

Natural sines, tangents, &c., defined ' 12 

Table XIV •. 243, 256 

Needle, magnetic, to adjust 41 

to re-magnetize 44 

hints as to management 44 

bearings should always be noted 82 

North Star, to establish meridian by 45 

times of meridian passage, Table 1 148 

extreme elongations of, Table 1 14S 

azimuths and natural tangents, Table II ' . . . . 150 



INDEX. 343 

PAGB 

Obstacles in the field to vision 73 

to measnrement 74 

Ordinates of circular curves, to find 58, 59 

of parabola, to find 36 

of a r curve, Table X 155 

Parabola, ordinates of . 36, 59 

Plane trigonometry 12 

Powers and roots of numbers by logarithms .......... 7 

Propositions, general, in trigonometry * 15 

Protractor for curves described 84 

how to make 85 

Rails, table of spring for trackmen 143 

Radius of a curve, how to find . . . . . . . . . . . . . 52, 54, 56 

of a turnout curve 129 

plain rule for, on curves 133 

for narrow gauges ' .' . . . . . . . 134 

Radii and their logarithms. Table IX. . ...".'.'.'. . . . . 155 

Records, forms for ......"........ 81 

Refraction and curvature, correction for 28 

!^eversed curves. See Curves. 

Rise per mile of various grades, Table XVII. 332 

Rod, levelling . 28 

how to read 42 

Rodman, duties of . 83 

Roods and perches in decimals of an acre. Table tn 151 

Roots and powers of numbers by logarithms ......... 7 

Senior assistant, duties of 80 

equipment for 81 

Sines defined . , . 12 

Shadows, to fix true north by 44 

Slopes for topography, Table XV 268 

Slopeman, duties of 84 

Slope stakes, to set 30 

for earth excavation 31 

for embankment . . .... . 33 

for hillsides and rock 85 

field record of work 84 

Spring of rails, table for trackmen 143 

Squares, cubes, and roots of numbers. Table XI. 161 

Supplement of an angle 12 

Survey, form for record 81 

to facilitate 82 

Switch-rails, angles of 130 

tables of 135,138 

Tangent, or apex distance of curve, to find 52, 54 



344 INDEX. 

PAGE 

Tangent of a 1° curve, Table XVI 269 

to curve from a fixed point, how to locate 108 

to two cui'ves on the ground, how to locate 109 

Tangential angles and distances explained 50 

how to find 57, 58, 64 

Thermometer, levelling by 29 

Track problems 115 

Trackmen's plain rules for finding frog distances 132, 133 

tables of turnouts 135,136 

plain rules for laying off turnouts with tape-measure and pins . 137 
, .crossings on straight lines and on curves ....... 139 

elevation of outer rail . 142 

instructions how to put in missing stakes on curves with tape- 

- ? measure 144 

table of curves and spring of rails .......... .143 

explanation of the trackmen's tables , . 144 

how to find the degree of a curve 145, 146 

Transit, adjustment of 40 

cross-hairs . . 24 

Transitman, duties of 82 

Triangles, solution of, — 

two angles and a side given 16 

two sides and an angle given 17 

three sides given 18 

Triangles, right-angled, solution of - 19 

Trigonometry, plane 12 

general propositions 15 

Turnouts. See Trackmen. 

Vernier explained ,..-.....,.- 42 

on transit 43 

Versed sines defined 12 

to calculate 54, 58 

of a r curve. Table XVi . 269 

Vertical curves, to calculate 36 

to project 39 



Tables: 

Ordinates of a 1° curve • 60 

For locating terminal curves 88 

Tangents between curves of contrary flexure ....... 89 

Turnouts, switch-rails straight . 135 

switch-rails curved 136 

Elevation of outer rail on curves 142 

Curves and spring of ra?ls . . . » 143 

I. Culminations and elongations of North Star 148 



INDEX. 345 

PAGE 

n. Azimuths of North Star, and their natural tangents „ , , 150 

in. Roods and perches in decimal parts of an acre 151 

IV. Decimals of an acre in one chain length of 100 feet, and of 

various widths 151 

V. Acres, roods, and perches in square feet , 152 

VI. Circular arcs to radius of 1 152 

VII. Feet in decimals of a mile 153 

VIII. Inches reduced to decimal parts of a foot 153 

IX. Radii and their logarithms, middle ordinates, and deflection 

distances 155 

X. Metric-curve table 159 

XI. Squares, cubes, roots, and reciprocals of numbers, from 1 

to 1,042 , 161 

XII. Logarithms of numbers from 1 to 10,000 179 

XIII. Logarithmic sines, cosines, tangents, and cotangents ... 197 

XIV. Natural sines and cosines 243 

Natural tangents and cotangents 256 

XV. Slopes for topography 268 

XVI. Functions of a 1° curve 269 

ivVn. Rise per mile of various grades , 388 

ADDENDA. 

Table of feet, inches and reciprocals of various track gauges .... 335 

Table of minutes of a degree expressed in decimals 335 

Condensed table of radii including short chords . 336 

To express gradients per cent, in angular measure 337 

Sound , 337 

American and French equivalents— Linear measure 337 

American and French equivalents— Square measure 338 

American and French equivalents— Cubic measure 338 



APR 1 1903 



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